Package 'distributions3' reference manual

Title:	Probability Distributions as S3 Objects
Description:	Tools to create and manipulate probability distributions using S3. Generics pdf(), cdf(), quantile(), and random() provide replacements for base R's d/p/q/r style functions. Functions and arguments have been named carefully to minimize confusion for students in intro stats courses. The documentation for each distribution contains detailed mathematical notes.
Authors:	Alex Hayes [aut, cre] , Ralph Moller-Trane [aut], Emil Hvitfeldt [ctb] , Daniel Jordan [aut], Paul Northrop [aut], Moritz N. Lang [aut] , Achim Zeileis [aut] , Bruna Wundervald [ctb], Alessandro Gasparini [ctb]
Maintainer:	Alex Hayes <[email protected]>
License:	MIT + file LICENSE
Version:	0.2.2.9000
Built:	2024-09-17 07:16:38 UTC
Source:	https://github.com/alexpghayes/distributions3

Utilities for `distributions3` objects

Description

Various utility functions to implement methods for distributions with a unified workflow, in particular to facilitate working with vectorized distributions3 objects. These are particularly useful in the computation of densities, probabilities, quantiles, and random samples when classical d/p/q/r functions are readily available for the distribution of interest.

Usage

apply_dpqr(d, FUN, at, elementwise = NULL, drop = TRUE, type = NULL, ...)

make_support(min, max, d, drop = TRUE)

make_positive_integer(n)
apply_dpqr(d, FUN, at, elementwise = NULL, drop = TRUE, type = NULL, ...)

make_support(min, max, d, drop = TRUE)

make_positive_integer(n)

Arguments

`d`	A `distributions3` object.
`FUN`	Function to be computed. Function should be of type `FUN(at, d)`, where `at` is the argument at which the function should be evaluated (e.g., a quantile, probability, or sample size) and `d` is a `distributions3` object.
`at`	Specification of values at which `FUN` should be evaluated, typically a numeric vector (e.g., of quantiles, probabilities, etc.) but possibly also a matrix or data frame.
`elementwise`	logical. Should each element of `d` only be evaluated at the corresponding element of `at` (`elementwise = TRUE`) or at all elements in `at` (`elementwise = FALSE`). Elementwise evaluation is only possible if the length of `d` and `at` is the same and in that case a vector of the same length is returned. Otherwise a matrix is returned. The default is to use `elementwise = TRUE` if possible, and otherwise `elementwise = FALSE`.
`drop`	logical. Should the result be simplified to a vector if possible (by dropping the dimension attribute)? If `FALSE` a matrix is always returned.
`type`	Character string used for naming, typically one of `"density"`, `"logLik"`, `"probability"`, `"quantile"`, and `"random"`. Note that the `"random"` case is processed differently internally in order to vectorize the random number generation more efficiently.
`...`	Arguments to be passed to `FUN`.
`min`, `max`	Numeric vectors. Minima and maxima of the supports of a `distributions3` object.
`n`	numeric. Number of observations for computing random draws. If `length(n) > 1`, the length is taken to be the number required (consistent with base R as, e.g., for `rnorm()`).

Examples

## Implementing a new distribution based on the provided utility functions
## Illustration: Gaussian distribution
## Note: Gaussian() is really just a copy of Normal() with a different class/distribution name


## Generator function for the distribution object.
Gaussian <- function(mu = 0, sigma = 1) {
  stopifnot(
    "parameter lengths do not match (only scalars are allowed to be recycled)" =
      length(mu) == length(sigma) | length(mu) == 1 | length(sigma) == 1
  )
  d <- data.frame(mu = mu, sigma = sigma)
  class(d) <- c("Gaussian", "distribution")
  d
}

## Set up a vector Y containing four Gaussian distributions:
Y <- Gaussian(mu = 1:4, sigma = c(1, 1, 2, 2))
Y

## Extract the underlying parameters:
as.matrix(Y)


## Extractor functions for moments of the distribution include
## mean(), variance(), skewness(), kurtosis().
## These can be typically be defined as functions of the list of parameters.
mean.Gaussian <- function(x, ...) {
  rlang::check_dots_used()
  setNames(x$mu, names(x))
}
## Analogously for other moments, see distributions3:::variance.Normal etc.

mean(Y)


## The support() method should return a matrix of "min" and "max" for the
## distribution. The make_support() function helps to set the right names and
## dimension.
support.Gaussian <- function(d, drop = TRUE, ...) {
  min <- rep(-Inf, length(d))
  max <- rep(Inf, length(d))
  make_support(min, max, d, drop = drop)
}

support(Y)


## Evaluating certain functions associated with the distribution, e.g.,
## pdf(), log_pdf(), cdf() quantile(), random(), etc. The apply_dpqr()
## function helps to call the typical d/p/q/r functions (like dnorm,
## pnorm, etc.) and set suitable names and dimension.
pdf.Gaussian <- function(d, x, elementwise = NULL, drop = TRUE, ...) {
  FUN <- function(at, d) dnorm(x = at, mean = d$mu, sd = d$sigma, ...)
  apply_dpqr(d = d, FUN = FUN, at = x, type = "density", elementwise = elementwise, drop = drop)
}

## Evaluate all densities at the same argument (returns vector):
pdf(Y, 0)

## Evaluate all densities at several arguments (returns matrix):
pdf(Y, c(0, 5))

## Evaluate each density at a different argument (returns vector):
pdf(Y, 4:1)

## Force evaluation of each density at a different argument (returns vector)
## or at all arguments (returns matrix):
pdf(Y, 4:1, elementwise = TRUE)
pdf(Y, 4:1, elementwise = FALSE)

## Drawing random() samples also uses apply_dpqr() with the argument
## n assured to be a positive integer.
random.Gaussian <- function(x, n = 1L, drop = TRUE, ...) {
  n <- make_positive_integer(n)
  if (n == 0L) {
    return(numeric(0L))
  }
  FUN <- function(at, d) rnorm(n = at, mean = d$mu, sd = d$sigma)
  apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}

## One random sample for each distribution (returns vector):
random(Y, 1)

## Several random samples for each distribution (returns matrix):
random(Y, 3)


## For further analogous methods see the "Normal" distribution provided
## in distributions3.
methods(class = "Normal")

## Implementing a new distribution based on the provided utility functions
## Illustration: Gaussian distribution
## Note: Gaussian() is really just a copy of Normal() with a different class/distribution name


## Generator function for the distribution object.
Gaussian <- function(mu = 0, sigma = 1) {
  stopifnot(
    "parameter lengths do not match (only scalars are allowed to be recycled)" =
      length(mu) == length(sigma) | length(mu) == 1 | length(sigma) == 1
  )
  d <- data.frame(mu = mu, sigma = sigma)
  class(d) <- c("Gaussian", "distribution")
  d
}

## Set up a vector Y containing four Gaussian distributions:
Y <- Gaussian(mu = 1:4, sigma = c(1, 1, 2, 2))
Y

## Extract the underlying parameters:
as.matrix(Y)


## Extractor functions for moments of the distribution include
## mean(), variance(), skewness(), kurtosis().
## These can be typically be defined as functions of the list of parameters.
mean.Gaussian <- function(x, ...) {
  rlang::check_dots_used()
  setNames(x$mu, names(x))
}
## Analogously for other moments, see distributions3:::variance.Normal etc.

mean(Y)


## The support() method should return a matrix of "min" and "max" for the
## distribution. The make_support() function helps to set the right names and
## dimension.
support.Gaussian <- function(d, drop = TRUE, ...) {
  min <- rep(-Inf, length(d))
  max <- rep(Inf, length(d))
  make_support(min, max, d, drop = drop)
}

support(Y)


## Evaluating certain functions associated with the distribution, e.g.,
## pdf(), log_pdf(), cdf() quantile(), random(), etc. The apply_dpqr()
## function helps to call the typical d/p/q/r functions (like dnorm,
## pnorm, etc.) and set suitable names and dimension.
pdf.Gaussian <- function(d, x, elementwise = NULL, drop = TRUE, ...) {
  FUN <- function(at, d) dnorm(x = at, mean = d$mu, sd = d$sigma, ...)
  apply_dpqr(d = d, FUN = FUN, at = x, type = "density", elementwise = elementwise, drop = drop)
}

## Evaluate all densities at the same argument (returns vector):
pdf(Y, 0)

## Evaluate all densities at several arguments (returns matrix):
pdf(Y, c(0, 5))

## Evaluate each density at a different argument (returns vector):
pdf(Y, 4:1)

## Force evaluation of each density at a different argument (returns vector)
## or at all arguments (returns matrix):
pdf(Y, 4:1, elementwise = TRUE)
pdf(Y, 4:1, elementwise = FALSE)

## Drawing random() samples also uses apply_dpqr() with the argument
## n assured to be a positive integer.
random.Gaussian <- function(x, n = 1L, drop = TRUE, ...) {
  n <- make_positive_integer(n)
  if (n == 0L) {
    return(numeric(0L))
  }
  FUN <- function(at, d) rnorm(n = at, mean = d$mu, sd = d$sigma)
  apply_dpqr(d = x, FUN = FUN, at = n, type = "random", drop = drop)
}

## One random sample for each distribution (returns vector):
random(Y, 1)

## Several random samples for each distribution (returns matrix):
random(Y, 3)


## For further analogous methods see the "Normal" distribution provided
## in distributions3.
methods(class = "Normal")

Create a Bernoulli distribution

Description

Bernoulli distributions are used to represent events like coin flips when there is single trial that is either successful or unsuccessful. The Bernoulli distribution is a special case of the Binomial() distribution with n = 1.

Usage

Bernoulli(p = 0.5)
Bernoulli(p = 0.5)

Arguments

`p`	The success probability for the distribution. `p` can be any value in `⁠[0, 1]⁠`, and defaults to `0.5`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a Bernoulli random variable with parameter p = $p$ . Some textbooks also define $q = 1 - p$ , or use $\pi$ instead of $p$ .

The Bernoulli probability distribution is widely used to model binary variables, such as 'failure' and 'success'. The most typical example is the flip of a coin, when $p$ is thought as the probability of flipping a head, and $q = 1 - p$ is the probability of flipping a tail.

Support: $\{0, 1\}$

Mean: $p$

Variance: $p \cdot (1 - p) = p \cdot q$

Probability mass function (p.m.f):

$P(X = x) = p^x (1 - p)^{1-x} = p^x q^{1-x}$

Cumulative distribution function (c.d.f):

$P(X \le x) = \left \{ \begin{array}{ll} 0 & x < 0 \\ 1 - p & 0 \leq x < 1 \\ 1 & x \geq 1 \end{array} \right.$

Moment generating function (m.g.f):

$E(e^{tX}) = (1 - p) + p e^t$

Value

A Bernoulli object.

Examples


set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Beta distribution

Description

Create a Beta distribution

Usage

Beta(alpha = 1, beta = 1)
Beta(alpha = 1, beta = 1)

Arguments

`alpha`	The alpha parameter. `alpha` can be any value strictly greater than zero. Defaults to `1`.
`beta`	The beta parameter. `beta` can be any value strictly greater than zero. Defaults to `1`.

Value

A beta object.

Examples


set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Binomial distribution

Description

Binomial distributions are used to represent situations can that can be thought as the result of $n$ Bernoulli experiments (here the $n$ is defined as the size of the experiment). The classical example is $n$ independent coin flips, where each coin flip has probability p of success. In this case, the individual probability of flipping heads or tails is given by the Bernoulli(p) distribution, and the probability of having $x$ equal results ( $x$ heads, for example), in $n$ trials is given by the Binomial(n, p) distribution. The equation of the Binomial distribution is directly derived from the equation of the Bernoulli distribution.

Usage

Binomial(size, p = 0.5)
Binomial(size, p = 0.5)

Arguments

`size`	The number of trials. Must be an integer greater than or equal to one. When `size = 1L`, the Binomial distribution reduces to the bernoulli distribution. Often called `n` in textbooks.
`p`	The success probability for a given trial. `p` can be any value in `⁠[0, 1]⁠`, and defaults to `0.5`.

Details

The Binomial distribution comes up when you are interested in the portion of people who do a thing. The Binomial distribution also comes up in the sign test, sometimes called the Binomial test (see stats::binom.test()), where you may need the Binomial C.D.F. to compute p-values.

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a Binomial random variable with parameter size = $n$ and p = $p$ . Some textbooks define $q = 1 - p$ , or called $\pi$ instead of $p$ .

Support: $\{0, 1, 2, ..., n\}$

Mean: $np$

Variance: $np \cdot (1 - p) = np \cdot q$

Probability mass function (p.m.f):

$P(X = k) = {n \choose k} p^k (1 - p)^{n-k}$

Cumulative distribution function (c.d.f):

$P(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} {n \choose i} p^i (1 - p)^{n-i}$

Moment generating function (m.g.f):

$E(e^{tX}) = (1 - p + p e^t)^n$

Value

A Binomial object.

Examples


set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create a Categorical distribution

Description

Create a Categorical distribution

Usage

Categorical(outcomes, p = NULL)
Categorical(outcomes, p = NULL)

Arguments

`outcomes`	A vector specifying the elements in the sample space. Can be numeric, factor, character, or logical.
`p`	A vector of success probabilities for each outcome. Each element of `p` can be any positive value – the vector gets normalized internally. Defaults to `NULL`, in which case the distribution is assumed to be uniform.

Value

A Categorical object.

Examples


set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)
set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Create a Cauchy distribution

Description

Note that the Cauchy distribution is the student's t distribution with one degree of freedom. The Cauchy distribution does not have a well defined mean or variance. Cauchy distributions often appear as priors in Bayesian contexts due to their heavy tails.

Usage

Cauchy(location = 0, scale = 1)
Cauchy(location = 0, scale = 1)

Arguments

`location`	The location parameter. Can be any real number. Defaults to `0`.
`scale`	The scale parameter. Must be greater than zero (?). Defaults to `1`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Cauchy variable with mean ⁠location =⁠ $x_0$ and scale = $\gamma$ .

Support: $R$ , the set of all real numbers

Mean: Undefined.

Variance: Undefined.

Probability density function (p.d.f):

$f(x) = \frac{1}{\pi \gamma \left[1 + \left(\frac{x - x_0}{\gamma} \right)^2 \right]}$

Cumulative distribution function (c.d.f):

$F(t) = \frac{1}{\pi} \arctan \left( \frac{t - x_0}{\gamma} \right) + \frac{1}{2}$

Moment generating function (m.g.f):

Does not exist.

Value

A Cauchy object.

Examples


set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a probability distribution

Description

Generic function for computing probabilities from distribution objects based on the cumulative distribution function (CDF).

Usage

cdf(d, x, drop = TRUE, ...)
cdf(d, x, drop = TRUE, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Probabilities corresponding to the vector x.

Examples

## distribution object
X <- Normal()
## probabilities from CDF
cdf(X, c(1, 2, 3, 4, 5))
## distribution object
X <- Normal()
## probabilities from CDF
cdf(X, c(1, 2, 3, 4, 5))

Evaluate the cumulative distribution function of a Bernoulli distribution

Description

Evaluate the cumulative distribution function of a Bernoulli distribution

Usage

## S3 method for class 'Bernoulli'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Bernoulli'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Bernoulli` object created by a call to `Bernoulli()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(x) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(x) columns containing all possible combinations.

Examples


set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Beta distribution

Description

Evaluate the cumulative distribution function of a Beta distribution

Usage

## S3 method for class 'Beta'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Beta'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Beta` object created by a call to `Beta()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pbeta`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Binomial distribution

Description

Evaluate the cumulative distribution function of a Binomial distribution

Usage

## S3 method for class 'Binomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Binomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Binomial` object created by a call to `Binomial()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Categorical distribution

Description

Evaluate the cumulative distribution function of a Categorical distribution

Usage

## S3 method for class 'Categorical'
cdf(d, x, ...)
## S3 method for class 'Categorical'
cdf(d, x, ...)

Arguments

`d`	A `Categorical` object created by a call to `Categorical()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector of probabilities, one for each element of x.

Examples


set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)
set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Evaluate the cumulative distribution function of a Cauchy distribution

Description

Evaluate the cumulative distribution function of a Cauchy distribution

Usage

## S3 method for class 'Cauchy'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Cauchy'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Cauchy` object created by a call to `Cauchy()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pcauchy`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a chi square distribution

Description

Evaluate the cumulative distribution function of a chi square distribution

Usage

## S3 method for class 'ChiSquare'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ChiSquare'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ChiSquare` object created by a call to `ChiSquare()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pchisq`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of an Erlang distribution

Description

Evaluate the cumulative distribution function of an Erlang distribution

Usage

## S3 method for class 'Erlang'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Erlang'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	An `Erlang` object created by a call to `Erlang()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgamma`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of an Exponential distribution

Description

Evaluate the cumulative distribution function of an Exponential distribution

Usage

## S3 method for class 'Exponential'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Exponential'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	An `Exponential` object created by a call to `Exponential()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pexp`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of an F distribution

Description

Evaluate the cumulative distribution function of an F distribution

Usage

## S3 method for class 'FisherF'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'FisherF'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `FisherF` object created by a call to `FisherF()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pf`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Frechet distribution

Description

Evaluate the cumulative distribution function of a Frechet distribution

Usage

## S3 method for class 'Frechet'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Frechet'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Frechet` object created by a call to `Frechet()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Gamma distribution

Description

Evaluate the cumulative distribution function of a Gamma distribution

Usage

## S3 method for class 'Gamma'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Gamma'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Gamma` object created by a call to `Gamma()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgamma`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Geometric distribution

Description

Evaluate the cumulative distribution function of a Geometric distribution

Usage

## S3 method for class 'Geometric'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Geometric'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Geometric` object created by a call to `Geometric()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgeom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a GEV distribution

Description

Evaluate the cumulative distribution function of a GEV distribution

Usage

## S3 method for class 'GEV'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'GEV'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `GEV` object created by a call to `GEV()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a GP distribution

Description

Evaluate the cumulative distribution function of a GP distribution

Usage

## S3 method for class 'GP'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'GP'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `GP` object created by a call to `GP()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgp`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Gumbel distribution

Description

Evaluate the cumulative distribution function of a Gumbel distribution

Usage

## S3 method for class 'Gumbel'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Gumbel'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Gumbel` object created by a call to `Gumbel()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a hurdle negative binomial distribution

Description

Evaluate the cumulative distribution function of a hurdle negative binomial distribution

Usage

## S3 method for class 'HurdleNegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HurdleNegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `HurdleNegativeBinomial` object created by a call to `HurdleNegativeBinomial()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `phnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a hurdle Poisson distribution

Description

Evaluate the cumulative distribution function of a hurdle Poisson distribution

Usage

## S3 method for class 'HurdlePoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HurdlePoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `HurdlePoisson` object created by a call to `HurdlePoisson()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `phpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a HyperGeometric distribution

Description

Evaluate the cumulative distribution function of a HyperGeometric distribution

Usage

## S3 method for class 'HyperGeometric'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HyperGeometric'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `HyperGeometric` object created by a call to `HyperGeometric()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `phyper`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a Logistic distribution

Description

Evaluate the cumulative distribution function of a Logistic distribution

Usage

## S3 method for class 'Logistic'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Logistic'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Logistic` object created by a call to `Logistic()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `plogis`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a LogNormal distribution

Description

Evaluate the cumulative distribution function of a LogNormal distribution

Usage

## S3 method for class 'LogNormal'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'LogNormal'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `LogNormal` object created by a call to `LogNormal()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `plnorm`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a negative binomial distribution

Description

Evaluate the cumulative distribution function of a negative binomial distribution

Usage

## S3 method for class 'NegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'NegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `NegativeBinomial` object created by a call to `NegativeBinomial()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)
set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Evaluate the cumulative distribution function of a Normal distribution

Description

Evaluate the cumulative distribution function of a Normal distribution

Usage

## S3 method for class 'Normal'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Normal'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Normal` object created by a call to `Normal()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pnorm`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a Poisson distribution

Description

Evaluate the cumulative distribution function of a Poisson distribution

Usage

## S3 method for class 'Poisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Poisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Poisson` object created by a call to `Poisson()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `ppois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the cumulative distribution function of a PoissonBinomial distribution

Description

Evaluate the cumulative distribution function of a PoissonBinomial distribution

Usage

## S3 method for class 'PoissonBinomial'
cdf(
  d,
  x,
  drop = TRUE,
  elementwise = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  verbose = TRUE,
  ...
)
## S3 method for class 'PoissonBinomial'
cdf(
  d,
  x,
  drop = TRUE,
  elementwise = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  verbose = TRUE,
  ...
)

Arguments

`d`	A `PoissonBinomial` object created by a call to `PoissonBinomial()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`lower.tail`, `log.p`, `...`	Arguments to be passed to `ppbinom` or `pnorm`, respectively.
`verbose`	logical. Should a warning be issued if the normal approximation is applied because the PoissonBinomial package is not installed?

Value

Examples


set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])
set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Evaluate the cumulative distribution function of an RevWeibull distribution

Description

Evaluate the cumulative distribution function of an RevWeibull distribution

Usage

## S3 method for class 'RevWeibull'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'RevWeibull'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `RevWeibull` object created by a call to `RevWeibull()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a StudentsT distribution

Description

Evaluate the cumulative distribution function of a StudentsT distribution

Usage

## S3 method for class 'StudentsT'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'StudentsT'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `StudentsT` object created by a call to `StudentsT()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pt`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Evaluate the cumulative distribution function of a Tukey distribution

Description

Evaluate the cumulative distribution function of a Tukey distribution

Usage

## S3 method for class 'Tukey'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Tukey'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Tukey` distribution created by a call to `Tukey()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `ptukey`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a continuous Uniform distribution

Description

Evaluate the cumulative distribution function of a continuous Uniform distribution

Usage

## S3 method for class 'Uniform'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Uniform'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Uniform` object created by a call to `Uniform()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `punif`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the cumulative distribution function of a Weibull distribution

Description

Evaluate the cumulative distribution function of a Weibull distribution

Usage

## S3 method for class 'Weibull'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Weibull'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Weibull` object created by a call to `Weibull()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pweibull`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the cumulative distribution function of a zero-inflated negative binomial distribution

Description

Evaluate the cumulative distribution function of a zero-inflated negative binomial distribution

Usage

## S3 method for class 'ZINegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZINegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZINegativeBinomial` object created by a call to `ZINegativeBinomial()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pzinbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a zero-inflated Poisson distribution

Description

Evaluate the cumulative distribution function of a zero-inflated Poisson distribution

Usage

## S3 method for class 'ZIPoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZIPoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZIPoisson` object created by a call to `ZIPoisson()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pzipois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a zero-truncated negative binomial distribution

Description

Evaluate the cumulative distribution function of a zero-truncated negative binomial distribution

Usage

## S3 method for class 'ZTNegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZTNegativeBinomial'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZTNegativeBinomial` object created by a call to `ZTNegativeBinomial()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pztnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the cumulative distribution function of a zero-truncated Poisson distribution

Description

Evaluate the cumulative distribution function of a zero-truncated Poisson distribution

Usage

## S3 method for class 'ZTPoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZTPoisson'
cdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZTPoisson` object created by a call to `ZTPoisson()`.
`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `pztpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a Chi-Square distribution

Description

Chi-square distributions show up often in frequentist settings as the sampling distribution of test statistics, especially in maximum likelihood estimation settings.

Usage

ChiSquare(df)
ChiSquare(df)

Arguments

`df`	Degrees of freedom. Must be positive.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a $\chi^2$ random variable with df = $k$ .

Support: $R^+$ , the set of positive real numbers

Mean: $k$

Variance: $2k$

Probability density function (p.d.f):

$f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}$

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

$F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx$

but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard normal is sometimes called the "error function". The notation $\Phi(t)$ also stands for the c.d.f. of a standard normal evaluated at $t$ . Z-tables list the value of $\Phi(t)$ for various $t$ .

Moment generating function (m.g.f):

$E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}$

Value

A ChiSquare object.

Transformations

A squared standard Normal() distribution is equivalent to a $\chi^2_1$ distribution with one degree of freedom. The $\chi^2$ distribution is a special case of the Gamma() distribution with shape (TODO: check this) parameter equal to a half. Sums of $\chi^2$ distributions are also distributed as $\chi^2$ distributions, where the degrees of freedom of the contributing distributions get summed. The ratio of two $\chi^2$ distributions is a FisherF() distribution. The ratio of a Normal() and the square root of a scaled ChiSquare() is a StudentsT() distribution.

Examples


set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

The hurdle negative binomial distribution

Description

Density, distribution function, quantile function, and random generation for the zero-hurdle negative binomial distribution with parameters mu, theta (or size), and pi.

Usage

dhnbinom(x, mu, theta, size, pi, log = FALSE)

phnbinom(q, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

qhnbinom(p, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

rhnbinom(n, mu, theta, size, pi)
dhnbinom(x, mu, theta, size, pi, log = FALSE)

phnbinom(q, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

qhnbinom(p, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

rhnbinom(n, mu, theta, size, pi)

Arguments

`x`	vector of (non-negative integer) quantiles.
`mu`	vector of (non-negative) negative binomial location parameters.
`theta`, `size`	vector of (non-negative) negative binomial overdispersion parameters. Only `theta` or, equivalently, `size` may be specified.
`pi`	vector of zero-hurdle probabilities in the unit interval.
`log`, `log.p`	logical indicating whether probabilities p are given as log(p).
`q`	vector of quantiles.
`lower.tail`	logical indicating whether probabilities are $P[X \le x]$ (lower tail) or $P[X > x]$ (upper tail).
`p`	vector of probabilities.
`n`	number of random values to return.

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four hnbinom functions for the hurdle negative binomial distribution call the corresponding nbinom functions for the negative binomial distribution from base R internally.

Note, however, that the precision of qhnbinom for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

Examples

## theoretical probabilities for a hurdle negative binomial distribution
x <- 0:8
p <- dhnbinom(x, mu = 2.5, theta = 1, pi = 0.75)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rhnbinom(500, mu = 2.5, theta = 1, pi = 0.75)
hist(y, breaks = -1:max(y) + 0.5)

## theoretical probabilities for a hurdle negative binomial distribution
x <- 0:8
p <- dhnbinom(x, mu = 2.5, theta = 1, pi = 0.75)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rhnbinom(500, mu = 2.5, theta = 1, pi = 0.75)
hist(y, breaks = -1:max(y) + 0.5)

The hurdle Poisson distribution

Description

Density, distribution function, quantile function, and random generation for the zero-hurdle Poisson distribution with parameters lambda and pi.

Usage

dhpois(x, lambda, pi, log = FALSE)

phpois(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qhpois(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rhpois(n, lambda, pi)
dhpois(x, lambda, pi, log = FALSE)

phpois(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qhpois(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rhpois(n, lambda, pi)

Arguments

`x`	vector of (non-negative integer) quantiles.
`lambda`	vector of (non-negative) Poisson parameters.
`pi`	vector of zero-hurdle probabilities in the unit interval.
`log`, `log.p`	logical indicating whether probabilities p are given as log(p).
`q`	vector of quantiles.
`lower.tail`	logical indicating whether probabilities are $P[X \le x]$ (lower tail) or $P[X > x]$ (upper tail).
`p`	vector of probabilities.
`n`	number of random values to return.

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four hpois functions for the hurdle Poisson distribution call the corresponding pois functions for the Poisson distribution from base R internally.

Note, however, that the precision of qhpois for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

Examples

## theoretical probabilities for a hurdle Poisson distribution
x <- 0:8
p <- dhpois(x, lambda = 2.5, pi = 0.75)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rhpois(500, lambda = 2.5, pi = 0.75)
hist(y, breaks = -1:max(y) + 0.5)

## theoretical probabilities for a hurdle Poisson distribution
x <- 0:8
p <- dhpois(x, lambda = 2.5, pi = 0.75)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rhpois(500, lambda = 2.5, pi = 0.75)
hist(y, breaks = -1:max(y) + 0.5)

The zero-inflated negative binomial distribution

Description

Density, distribution function, quantile function, and random generation for the zero-inflated negative binomial distribution with parameters mu, theta (or size), and pi.

Usage

dzinbinom(x, mu, theta, size, pi, log = FALSE)

pzinbinom(q, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

qzinbinom(p, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

rzinbinom(n, mu, theta, size, pi)
dzinbinom(x, mu, theta, size, pi, log = FALSE)

pzinbinom(q, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

qzinbinom(p, mu, theta, size, pi, lower.tail = TRUE, log.p = FALSE)

rzinbinom(n, mu, theta, size, pi)

Arguments

`x`	vector of (non-negative integer) quantiles.
`mu`	vector of (non-negative) negative binomial location parameters.
`theta`, `size`	vector of (non-negative) negative binomial overdispersion parameters. Only `theta` or, equivalently, `size` may be specified.
`pi`	vector of zero-inflation probabilities in the unit interval.
`log`, `log.p`	logical indicating whether probabilities p are given as log(p).
`q`	vector of quantiles.
`lower.tail`	logical indicating whether probabilities are $P[X \le x]$ (lower tail) or $P[X > x]$ (upper tail).
`p`	vector of probabilities.
`n`	number of random values to return.

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four zinbinom functions for the zero-inflated negative binomial distribution call the corresponding nbinom functions for the negative binomial distribution from base R internally.

Note, however, that the precision of qzinbinom for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

Examples

## theoretical probabilities for a zero-inflated negative binomial distribution
x <- 0:8
p <- dzinbinom(x, mu = 2.5, theta = 1, pi = 0.25)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rzinbinom(500, mu = 2.5, theta = 1, pi = 0.25)
hist(y, breaks = -1:max(y) + 0.5)

## theoretical probabilities for a zero-inflated negative binomial distribution
x <- 0:8
p <- dzinbinom(x, mu = 2.5, theta = 1, pi = 0.25)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rzinbinom(500, mu = 2.5, theta = 1, pi = 0.25)
hist(y, breaks = -1:max(y) + 0.5)

The zero-inflated Poisson distribution

Description

Density, distribution function, quantile function, and random generation for the zero-inflated Poisson distribution with parameters lambda and pi.

Usage

dzipois(x, lambda, pi, log = FALSE)

pzipois(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qzipois(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rzipois(n, lambda, pi)
dzipois(x, lambda, pi, log = FALSE)

pzipois(q, lambda, pi, lower.tail = TRUE, log.p = FALSE)

qzipois(p, lambda, pi, lower.tail = TRUE, log.p = FALSE)

rzipois(n, lambda, pi)

Arguments

`x`	vector of (non-negative integer) quantiles.
`lambda`	vector of (non-negative) Poisson parameters.
`pi`	vector of zero-inflation probabilities in the unit interval.
`log`, `log.p`	logical indicating whether probabilities p are given as log(p).
`q`	vector of quantiles.
`lower.tail`	logical indicating whether probabilities are $P[X \le x]$ (lower tail) or $P[X > x]$ (upper tail).
`p`	vector of probabilities.
`n`	number of random values to return.

Details

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four zipois functions for the zero-inflated Poisson distribution call the corresponding pois functions for the Poisson distribution from base R internally.

Note, however, that the precision of qzipois for very large probabilities (close to 1) is limited because the probabilities are internally handled in levels and not in logs (even if log.p = TRUE).

Examples

## theoretical probabilities for a zero-inflated Poisson distribution
x <- 0:8
p <- dzipois(x, lambda = 2.5, pi = 0.25)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rzipois(500, lambda = 2.5, pi = 0.25)
hist(y, breaks = -1:max(y) + 0.5)

## theoretical probabilities for a zero-inflated Poisson distribution
x <- 0:8
p <- dzipois(x, lambda = 2.5, pi = 0.25)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rzipois(500, lambda = 2.5, pi = 0.25)
hist(y, breaks = -1:max(y) + 0.5)

The zero-truncated negative binomial distribution

Description

Density, distribution function, quantile function, and random generation for the zero-truncated negative binomial distribution with parameters mu and theta (or size).

Usage

dztnbinom(x, mu, theta, size, log = FALSE)

pztnbinom(q, mu, theta, size, lower.tail = TRUE, log.p = FALSE)

qztnbinom(p, mu, theta, size, lower.tail = TRUE, log.p = FALSE)

rztnbinom(n, mu, theta, size)
dztnbinom(x, mu, theta, size, log = FALSE)

pztnbinom(q, mu, theta, size, lower.tail = TRUE, log.p = FALSE)

qztnbinom(p, mu, theta, size, lower.tail = TRUE, log.p = FALSE)

rztnbinom(n, mu, theta, size)

Arguments

`x`	vector of (non-negative integer) quantiles.
`mu`	vector of (non-negative) negative binomial location parameters.
`theta`, `size`	vector of (non-negative) negative binomial overdispersion parameters. Only `theta` or, equivalently, `size` may be specified.
`log`, `log.p`	logical indicating whether probabilities p are given as log(p).
`q`	vector of quantiles.
`lower.tail`	logical indicating whether probabilities are $P[X \le x]$ (lower tail) or $P[X > x]$ (upper tail).
`p`	vector of probabilities.
`n`	number of random values to return.

Details

The negative binomial distribution left-truncated at zero (or zero-truncated negative binomial for short) is the distribution obtained, when considering a negative binomial variable Y conditional on Y being greater than zero.

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four ztnbinom functions for the zero-truncated negative binomial distribution call the corresponding nbinom functions for the negative binomial distribution from base R internally.

Examples

## theoretical probabilities for a zero-truncated negative binomial distribution
x <- 0:8
p <- dztnbinom(x, mu = 2.5, theta = 1)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rztnbinom(500, mu = 2.5, theta = 1)
hist(y, breaks = -1:max(y) + 0.5)

## theoretical probabilities for a zero-truncated negative binomial distribution
x <- 0:8
p <- dztnbinom(x, mu = 2.5, theta = 1)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rztnbinom(500, mu = 2.5, theta = 1)
hist(y, breaks = -1:max(y) + 0.5)

The zero-truncated Poisson distribution

Description

Density, distribution function, quantile function, and random generation for the zero-truncated Poisson distribution with parameter lambda.

Usage

dztpois(x, lambda, log = FALSE)

pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE)

qztpois(p, lambda, lower.tail = TRUE, log.p = FALSE)

rztpois(n, lambda)
dztpois(x, lambda, log = FALSE)

pztpois(q, lambda, lower.tail = TRUE, log.p = FALSE)

qztpois(p, lambda, lower.tail = TRUE, log.p = FALSE)

rztpois(n, lambda)

Arguments

`x`	vector of (non-negative integer) quantiles.
`lambda`	vector of (non-negative) Poisson parameters.
`log`, `log.p`	logical indicating whether probabilities p are given as log(p).
`q`	vector of quantiles.
`lower.tail`	logical indicating whether probabilities are $P[X \le x]$ (lower tail) or $P[X > x]$ (upper tail).
`p`	vector of probabilities.
`n`	number of random values to return.

Details

The Poisson distribution left-truncated at zero (or zero-truncated Poisson for short) is the distribution obtained, when considering a Poisson variable Y conditional on Y being greater than zero.

All functions follow the usual conventions of d/p/q/r functions in base R. In particular, all four ztpois functions for the zero-truncated Poisson distribution call the corresponding pois functions for the Poisson distribution from base R internally.

Examples

## theoretical probabilities for a zero-truncated Poisson distribution
x <- 0:8
p <- dztpois(x, lambda = 2.5)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rztpois(500, lambda = 2.5)
hist(y, breaks = -1:max(y) + 0.5)

## theoretical probabilities for a zero-truncated Poisson distribution
x <- 0:8
p <- dztpois(x, lambda = 2.5)
plot(x, p, type = "h", lwd = 2)

## corresponding empirical frequencies from a simulated sample
set.seed(0)
y <- rztpois(500, lambda = 2.5)
hist(y, breaks = -1:max(y) + 0.5)

Create an Erlang distribution

Description

The Erlang distribution is a two-parameter family of continuous probability distributions with support $x \in [0,\infty)$ . The two parameters are a positive integer shape parameter $k$ and a positive real rate parameter $\lambda$ . The Erlang distribution with shape parameter $k = 1$ simplifies to the exponential distribution, and it is a special case of the gamma distribution. It corresponds to a sum of $k$ independent exponential variables with mean $1 / \lambda$ each.

Usage

Erlang(k, lambda)
Erlang(k, lambda)

Arguments

`k`	The shape parameter. Can be any positive integer number.
`lambda`	The rate parameter. Can be any positive number.

Value

An Erlang object.

Examples


set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create an Exponential distribution

Description

Exponential distributions are frequently used for modeling the amount of time that passes until a specific event occurs. For example, exponential distributions could be used to model the time between two earthquakes, the amount of delay between internet packets, or the amount of time a piece of machinery can run before needing repair.

Usage

Exponential(rate = 1)
Exponential(rate = 1)

Arguments

rate

The rate parameter, written $\lambda$ in textbooks. Can be any positive number. Defaults to 1.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be an Exponential random variable with rate parameter rate = $\lambda$ .

Support: $x \in (0, \infty)$

Mean: $\frac{1}{\lambda}$

Variance: $\frac{1}{\lambda^2}$

Probability density function (p.d.f):

$f(x) = \lambda e^{-\lambda x}$

Cumulative distribution function (c.d.f):

$F(x) = 1 - e^{-\lambda x}$

Moment generating function (m.g.f):

$\frac{\lambda}{\lambda - t}, for t < \lambda$

Value

An Exponential object.

Examples


set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Goals scored in all 2018 FIFA World Cup matches

Description

Data from all 64 matches in the 2018 FIFA World Cup along with predicted ability differences based on bookmakers odds.

Usage

data("FIFA2018", package = "distributions3")
data("FIFA2018", package = "distributions3")

Format

A data frame with 128 rows and 7 columns.

goals: integer. Number of goals scored in normal time (90 minutes), \ i.e., excluding potential extra time or penalties in knockout matches.
team: character. 3-letter FIFA code for the team.
match: integer. Match ID ranging from 1 (opening match) to 64 (final).
type: factor. Type of match for groups A to H, round of 16 (R16), quarter final, semi-final, match for 3rd place, and final.
stage: factor. Group vs. knockout tournament stage.
logability: numeric. Estimated log-ability for each team based on bookmaker consensus model.
difference: numeric. Difference in estimated log-abilities between a team and its opponent in each match.

Details

To investigate the number of goals scored per match in the 2018 FIFA World Cup, FIFA2018 provides two rows, one for each team, for each of the matches during the tournament. In addition some basic meta-information for the matches (an ID, team name abbreviations, type of match, group vs. knockout stage), information on the estimated log-ability for each team is provided. These have been estimated by Zeileis et al. (2018) prior to the start of the tournament (2018-05-20) based on quoted odds from 26 online bookmakers using the bookmaker consensus model of Leitner et al. (2010). The difference in log-ability between a team and its opponent is a useful predictor for the number of goals scored.

To model the data a basic Poisson regression model provides a good fit. This treats the number of goals by the two teams as independent given the ability difference which is a reasonable assumption in this data set.

Source

The goals for each match have been obtained from Wikipedia (https://en.wikipedia.org/wiki/2018_FIFA_World_Cup) and the log-abilities from Zeileis et al. (2018) based on quoted odds from Oddschecker.com and Bwin.com.

References

Leitner C, Zeileis A, Hornik K (2010). Forecasting Sports Tournaments by Ratings of (Prob)abilities: A Comparison for the EURO 2008. International Journal of Forecasting, 26(3), 471-481. doi:10.1016/j.ijforecast.2009.10.001

Zeileis A, Leitner C, Hornik K (2018). Probabilistic Forecasts for the 2018 FIFA World Cup Based on the Bookmaker Consensus Model. Working Paper 2018-09, Working Papers in Economics and Statistics, Research Platform Empirical and Experimental Economics, University of Innsbruck. https://EconPapers.RePEc.org/RePEc:inn:wpaper:2018-09

Examples

## load data
data("FIFA2018", package = "distributions3")

## observed relative frequencies of goals in all matches
obsrvd <- prop.table(table(FIFA2018$goals))

## expected probabilities assuming a simple Poisson model,
## using the average number of goals across all teams/matches
## as the point estimate for the mean (lambda) of the distribution
p_const <- Poisson(lambda = mean(FIFA2018$goals))
p_const
expctd <- pdf(p_const, 0:6)

## comparison: observed vs. expected frequencies
## frequencies for 3 and 4 goals are slightly overfitted
## while 5 and 6 goals are slightly underfitted
cbind("observed" = obsrvd, "expected" = expctd)

## instead of fitting the same average Poisson model to all
## teams/matches, take ability differences into account
m <- glm(goals ~ difference, data = FIFA2018, family = poisson)
summary(m)
## when the ratio of abilities increases by 1 percent, the
## expected number of goals increases by around 0.4 percent

## this yields a different predicted Poisson distribution for
## each team/match
p_reg <- Poisson(lambda = fitted(m))
head(p_reg)

## as an illustration, the following goal distributions
## were expected for the final (that France won 4-2 against Croatia)
p_final <- tail(p_reg, 2)
p_final
pdf(p_final, 0:6)
## clearly France was expected to score more goals than Croatia
## but both teams scored more goals than expected, albeit not unlikely many

## assuming independence of the number of goals scored, obtain
## table of possible match results (after normal time), along with
## overall probabilities of win/draw/lose
res <- outer(pdf(p_final[1], 0:6), pdf(p_final[2], 0:6))
sum(res[lower.tri(res)]) ## France wins
sum(diag(res))           ## draw
sum(res[upper.tri(res)]) ## France loses

## update expected frequencies table based on regression model
expctd <- pdf(p_reg, 0:6)
head(expctd)
expctd <- colMeans(expctd)
cbind("observed" = obsrvd, "expected" = expctd)
## load data
data("FIFA2018", package = "distributions3")

## observed relative frequencies of goals in all matches
obsrvd <- prop.table(table(FIFA2018$goals))

## expected probabilities assuming a simple Poisson model,
## using the average number of goals across all teams/matches
## as the point estimate for the mean (lambda) of the distribution
p_const <- Poisson(lambda = mean(FIFA2018$goals))
p_const
expctd <- pdf(p_const, 0:6)

## comparison: observed vs. expected frequencies
## frequencies for 3 and 4 goals are slightly overfitted
## while 5 and 6 goals are slightly underfitted
cbind("observed" = obsrvd, "expected" = expctd)

## instead of fitting the same average Poisson model to all
## teams/matches, take ability differences into account
m <- glm(goals ~ difference, data = FIFA2018, family = poisson)
summary(m)
## when the ratio of abilities increases by 1 percent, the
## expected number of goals increases by around 0.4 percent

## this yields a different predicted Poisson distribution for
## each team/match
p_reg <- Poisson(lambda = fitted(m))
head(p_reg)

## as an illustration, the following goal distributions
## were expected for the final (that France won 4-2 against Croatia)
p_final <- tail(p_reg, 2)
p_final
pdf(p_final, 0:6)
## clearly France was expected to score more goals than Croatia
## but both teams scored more goals than expected, albeit not unlikely many

## assuming independence of the number of goals scored, obtain
## table of possible match results (after normal time), along with
## overall probabilities of win/draw/lose
res <- outer(pdf(p_final[1], 0:6), pdf(p_final[2], 0:6))
sum(res[lower.tri(res)]) ## France wins
sum(diag(res))           ## draw
sum(res[upper.tri(res)]) ## France loses

## update expected frequencies table based on regression model
expctd <- pdf(p_reg, 0:6)
head(expctd)
expctd <- colMeans(expctd)
cbind("observed" = obsrvd, "expected" = expctd)

Create an F distribution

Description

Create an F distribution

Usage

FisherF(df1, df2, lambda = 0)
FisherF(df1, df2, lambda = 0)

Arguments

`df1`	Numerator degrees of freedom. Can be any positive number.
`df2`	Denominator degrees of freedom. Can be any positive number.
`lambda`	Non-centrality parameter. Can be any positive number. Defaults to `0`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

TODO

Value

A FisherF object.

Examples


set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Fit a distribution to data

Description

Generic function for fitting maximum-likelihood estimates (MLEs) of a distribution based on empirical data.

Usage

fit_mle(d, x, ...)
fit_mle(d, x, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`x`	A vector of data to compute the likelihood.
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A distribution (the same kind as d) where the parameters are the MLE estimates based on x.

Examples

X <- Normal()
fit_mle(X, c(-1, 0, 0, 0, 3))
X <- Normal()
fit_mle(X, c(-1, 0, 0, 0, 3))

Fit a Bernoulli distribution to data

Description

Fit a Bernoulli distribution to data

Usage

## S3 method for class 'Bernoulli'
fit_mle(d, x, ...)
## S3 method for class 'Bernoulli'
fit_mle(d, x, ...)

Arguments

`d`	A `Bernoulli` object.
`x`	A vector of zeroes and ones.
`...`	Unused.

Value

a Bernoulli object

Fit a Binomial distribution to data

Description

The fit distribution will inherit the same size parameter as the Binomial object passed.

Usage

## S3 method for class 'Binomial'
fit_mle(d, x, ...)
## S3 method for class 'Binomial'
fit_mle(d, x, ...)

Arguments

`d`	A `Binomial` object.
`x`	A vector of zeroes and ones.
`...`	Unused.

Value

a Binomial object

Fit an Exponential distribution to data

Description

Fit an Exponential distribution to data

Usage

## S3 method for class 'Exponential'
fit_mle(d, x, ...)
## S3 method for class 'Exponential'
fit_mle(d, x, ...)

Arguments

`d`	An `Exponential` object created by a call to `Exponential()`.
`x`	A vector of data.
`...`	Unused.

Value

An Exponential object.

Fit a Gamma distribution to data

Description

Fit a Gamma distribution to data

Usage

## S3 method for class 'Gamma'
fit_mle(d, x, ...)
## S3 method for class 'Gamma'
fit_mle(d, x, ...)

Arguments

`d`	A `Gamma` object created by a call to `Gamma()`.
`x`	A vector to fit the Gamma distribution to.
`...`	Unused.

Value

a Gamma object

Fit a Geometric distribution to data

Description

Fit a Geometric distribution to data

Usage

## S3 method for class 'Geometric'
fit_mle(d, x, ...)
## S3 method for class 'Geometric'
fit_mle(d, x, ...)

Arguments

`d`	A `Geometric` object.
`x`	A vector of zeroes and ones.
`...`	Unused.

Value

a Geometric object

Fit a Log Normal distribution to data

Description

Fit a Log Normal distribution to data

Usage

## S3 method for class 'LogNormal'
fit_mle(d, x, ...)
## S3 method for class 'LogNormal'
fit_mle(d, x, ...)

Arguments

`d`	A `LogNormal` object created by a call to `LogNormal()`.
`x`	A vector of data.
`...`	Unused.

Value

A LogNormal object.

Fit a Normal distribution to data

Description

Fit a Normal distribution to data

Usage

## S3 method for class 'Normal'
fit_mle(d, x, ...)
## S3 method for class 'Normal'
fit_mle(d, x, ...)

Arguments

`d`	A `Normal` object created by a call to `Normal()`.
`x`	A vector of data.
`...`	Unused.

Value

A Normal object.

Fit an Poisson distribution to data

Description

Fit an Poisson distribution to data

Usage

## S3 method for class 'Poisson'
fit_mle(d, x, ...)
## S3 method for class 'Poisson'
fit_mle(d, x, ...)

Arguments

`d`	An `Poisson` object created by a call to `Poisson()`.
`x`	A vector of data.
`...`	Unused.

Value

An Poisson object.

Create a Frechet distribution

Description

The Frechet distribution is a special case of the ⁠\link{GEV}⁠ distribution, obtained when the GEV shape parameter $\xi$ is positive. It may be referred to as a type II extreme value distribution.

Usage

Frechet(location = 0, scale = 1, shape = 1)
Frechet(location = 0, scale = 1, shape = 1)

Arguments

`location`	The location (minimum) parameter $m$ . `location` can be any real number. Defaults to `0`.
`scale`	The scale parameter $s$ . `scale` can be any positive number. Defaults to `1`.
`shape`	The shape parameter $\alpha$ . `shape` can be any positive number. Defaults to `1`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Frechet random variable with location parameter location = $m$ , scale parameter scale = $s$ , and shape parameter shape = $\alpha$ . A Frechet( $m, s, \alpha$ ) distribution is equivalent to a ⁠\link{GEV}⁠( $m + s, s / \alpha, 1 / \alpha$ ) distribution.

Support: $(m, \infty)$ .

Mean: $m + s\Gamma(1 - 1/\alpha)$ , for $\alpha > 1$ ; undefined otherwise.

Median: $m + s(\ln 2)^{-1/\alpha}$ .

Variance: $s^2 [\Gamma(1 - 2 / \alpha) - \Gamma(1 - 1 / \alpha)^2]$ for $\alpha > 2$ ; undefined otherwise.

Probability density function (p.d.f):

$f(x) = \alpha s ^ {-1} [(x - m) / s] ^ {-(1 + \alpha)}% \exp\{-[(x - m) / s] ^ {-\alpha} \}$

for $x > m$ . The p.d.f. is 0 for $x \leq m$ .

Cumulative distribution function (c.d.f):

$F(x) = \exp\{-[(x - m) / s] ^ {-\alpha} \}$

for $x > m$ . The c.d.f. is 0 for $x \leq m$ .

Value

A Frechet object.

Examples


set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Gamma distribution

Description

Several important distributions are special cases of the Gamma distribution. When the shape parameter is 1, the Gamma is an exponential distribution with parameter $1/\beta$ . When the $shape = n/2$ and $rate = 1/2$ , the Gamma is a equivalent to a chi squared distribution with n degrees of freedom. Moreover, if we have $X_1$ is $Gamma(\alpha_1, \beta)$ and $X_2$ is $Gamma(\alpha_2, \beta)$ , a function of these two variables of the form $\frac{X_1}{X_1 + X_2}$ $Beta(\alpha_1, \alpha_2)$ . This last property frequently appears in another distributions, and it has extensively been used in multivariate methods. More about the Gamma distribution will be added soon.

Usage

Gamma(shape, rate = 1)
Gamma(shape, rate = 1)

Arguments

`shape`	The shape parameter. Can be any positive number.
`rate`	The rate parameter. Can be any positive number. Defaults to `1`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a Gamma random variable with parameters shape = $\alpha$ and rate = $\beta$ .

Support: $x \in (0, \infty)$

Mean: $\frac{\alpha}{\beta}$

Variance: $\frac{\alpha}{\beta^2}$

Probability density function (p.m.f):

$f(x) = \frac{\beta^{\alpha}}{\Gamma(\alpha)} x^{\alpha - 1} e^{-\beta x}$

Cumulative distribution function (c.d.f):

$f(x) = \frac{\Gamma(\alpha, \beta x)}{\Gamma{\alpha}}$

Moment generating function (m.g.f):

$E(e^{tX}) = \Big(\frac{\beta}{ \beta - t}\Big)^{\alpha}, \thinspace t < \beta$

Value

A Gamma object.

Examples


set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create a Geometric distribution

Description

The Geometric distribution can be thought of as a generalization of the Bernoulli() distribution where we ask: "if I keep flipping a coin with probability p of heads, what is the probability I need $k$ flips before I get my first heads?" The Geometric distribution is a special case of Negative Binomial distribution.

Usage

Geometric(p = 0.5)
Geometric(p = 0.5)

Arguments

`p`	The success probability for the distribution. `p` can be any value in `⁠[0, 1]⁠`, and defaults to `0.5`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Geometric random variable with success probability p = $p$ . Note that there are multiple parameterizations of the Geometric distribution.

Support: 0 < p < 1, $x = 0, 1, \dots$

Mean: $\frac{1-p}{p}$

Variance: $\frac{1-p}{p^2}$

Probability mass function (p.m.f):

$P(X = x) = p(1-p)^x,$

Cumulative distribution function (c.d.f):

$P(X \le x) = 1 - (1-p)^{x+1}$

Moment generating function (m.g.f):

$E(e^{tX}) = \frac{pe^t}{1 - (1-p)e^t}$

Value

A Geometric object.

Examples


set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a Generalised Extreme Value (GEV) distribution

Description

The GEV distribution arises from the Extremal Types Theorem, which is rather like the Central Limit Theorem (see ⁠\link{Normal}⁠) but it relates to the maximum of $n$ i.i.d. random variables rather than to the sum. If, after a suitable linear rescaling, the distribution of this maximum tends to a non-degenerate limit as $n$ tends to infinity then this limit must be a GEV distribution. The requirement that the variables are independent can be relaxed substantially. Therefore, the GEV distribution is often used to model the maximum of a large number of random variables.

Usage

GEV(mu = 0, sigma = 1, xi = 0)
GEV(mu = 0, sigma = 1, xi = 0)

Arguments

`mu`	The location parameter, written $\mu$ in textbooks. `mu` can be any real number. Defaults to `0`.
`sigma`	The scale parameter, written $\sigma$ in textbooks. `sigma` can be any positive number. Defaults to `1`.
`xi`	The shape parameter, written $\xi$ in textbooks. `xi` can be any real number. Defaults to `0`, which corresponds to a Gumbel distribution.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a GEV random variable with location parameter mu = $\mu$ , scale parameter sigma = $\sigma$ and shape parameter xi = $\xi$ .

Support: $(-\infty, \mu - \sigma / \xi)$ for $\xi < 0$ ; $(\mu - \sigma / \xi, \infty)$ for $\xi > 0$ ; and $R$ , the set of all real numbers, for $\xi = 0$ .

Mean: $\mu + \sigma[\Gamma(1 - \xi) - 1]/\xi$ for $\xi < 1, \xi \neq 0$ ; $\mu + \sigma\gamma$ for $\xi = 0$ , where $\gamma$ is Euler's constant, approximately equal to 0.57722; undefined otherwise.

Median: $\mu + \sigma[(\ln 2) ^ {-\xi} - 1]/\xi$ for $\xi \neq 0$ ; $\mu - \sigma\ln(\ln 2)$ for $\xi = 0$ .

Variance: $\sigma^2 [\Gamma(1 - 2 \xi) - \Gamma(1 - \xi)^2] / \xi^2$ for $\xi < 1 / 2, \xi \neq 0$ ; $\sigma^2 \pi^2 / 6$ for $\xi = 0$ ; undefined otherwise.

Probability density function (p.d.f):

If $\xi \neq 0$ then

$f(x) = \sigma ^ {-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}% \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$

for $1 + \xi (x - \mu) / \sigma > 0$ . The p.d.f. is 0 outside the support.

In the $\xi = 0$ (Gumbel) special case

$f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]% \exp\{-\exp[-(x - \mu) / \sigma] \}$

for $x$ in $R$ , the set of all real numbers.

Cumulative distribution function (c.d.f):

If $\xi \neq 0$ then

$F(x) = \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$

for $1 + \xi (x - \mu) / \sigma > 0$ . The c.d.f. is 0 below the support and 1 above the support.

In the $\xi = 0$ (Gumbel) special case

$F(x) = \exp\{-\exp[-(x - \mu) / \sigma] \}$

for $x$ in $R$ , the set of all real numbers.

Value

A GEV object.

Examples


set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Generalised Pareto (GP) distribution

Description

The GP distribution has a link to the ⁠\link{GEV}⁠ distribution. Suppose that the maximum of $n$ i.i.d. random variables has approximately a GEV distribution. For a sufficiently large threshold $u$ , the conditional distribution of the amount (the threshold excess) by which a variable exceeds $u$ given that it exceeds $u$ has approximately a GP distribution. Therefore, the GP distribution is often used to model the threshold excesses of a high threshold $u$ . The requirement that the variables are independent can be relaxed substantially, but then exceedances of $u$ may cluster.

Usage

GP(mu = 0, sigma = 1, xi = 0)
GP(mu = 0, sigma = 1, xi = 0)

Arguments

`mu`	The location parameter, written $\mu$ in textbooks. `mu` can be any real number. Defaults to `0`.
`sigma`	The scale parameter, written $\sigma$ in textbooks. `sigma` can be any positive number. Defaults to `1`.
`xi`	The shape parameter, written $\xi$ in textbooks. `xi` can be any real number. Defaults to `0`, which corresponds to a Gumbel distribution.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a GP random variable with location parameter mu = $\mu$ , scale parameter sigma = $\sigma$ and shape parameter xi = $\xi$ .

Support: $[\mu, \mu - \sigma / \xi]$ for $\xi < 0$ ; $[\mu, \infty)$ for $\xi \geq 0$ .

Mean: $\mu + \sigma/(1 - \xi)$ for $\xi < 1$ ; undefined otherwise.

Median: $\mu + \sigma[2 ^ \xi - 1]/\xi$ for $\xi \neq 0$ ; $\mu + \sigma\ln 2$ for $\xi = 0$ .

Variance: $\sigma^2 / (1 - \xi)^2 (1 - 2\xi)$ for $\xi < 1 / 2$ ; undefined otherwise.

Probability density function (p.d.f):

If $\xi \neq 0$ then

$f(x) = \sigma^{-1} [1 + \xi (x - \mu) / \sigma] ^ {-(1 + 1/\xi)}$

for $1 + \xi (x - \mu) / \sigma > 0$ . The p.d.f. is 0 outside the support.

In the $\xi = 0$ special case

$f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]$

for $x$ in [ $\mu, \infty$ ). The p.d.f. is 0 outside the support.

Cumulative distribution function (c.d.f):

If $\xi \neq 0$ then

$F(x) = 1 - \exp\{-[1 + \xi (x - \mu) / \sigma] ^ {-1/\xi} \}$

for $1 + \xi (x - \mu) / \sigma > 0$ . The c.d.f. is 0 below the support and 1 above the support.

In the $\xi = 0$ special case

$F(x) = 1 - \exp[-(x - \mu) / \sigma] \}$

for $x$ in $R$ , the set of all real numbers.

Value

A GP object.

Examples


set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a Gumbel distribution

Description

The Gumbel distribution is a special case of the ⁠\link{GEV}⁠ distribution, obtained when the GEV shape parameter $\xi$ is equal to 0. It may be referred to as a type I extreme value distribution.

Usage

Gumbel(mu = 0, sigma = 1)
Gumbel(mu = 0, sigma = 1)

Arguments

`mu`	The location parameter, written $\mu$ in textbooks. `mu` can be any real number. Defaults to `0`.
`sigma`	The scale parameter, written $\sigma$ in textbooks. `sigma` can be any positive number. Defaults to `1`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Gumbel random variable with location parameter mu = $\mu$ , scale parameter sigma = $\sigma$ .

Support: $R$ , the set of all real numbers.

Mean: $\mu + \sigma\gamma$ , where $\gamma$ is Euler's constant, approximately equal to 0.57722.

Median: $\mu - \sigma\ln(\ln 2)$ .

Variance: $\sigma^2 \pi^2 / 6$ .

Probability density function (p.d.f):

$f(x) = \sigma ^ {-1} \exp[-(x - \mu) / \sigma]% \exp\{-\exp[-(x - \mu) / \sigma] \}$

for $x$ in $R$ , the set of all real numbers.

Cumulative distribution function (c.d.f):

In the $\xi = 0$ (Gumbel) special case

$F(x) = \exp\{-\exp[-(x - \mu) / \sigma] \}$

for $x$ in $R$ , the set of all real numbers.

Value

A Gumbel object.

Examples


set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Create a hurdle negative binomial distribution

Description

Hurdle negative binomial distributions are frequently used to model counts with overdispersion and many zero observations.

Usage

HurdleNegativeBinomial(mu, theta, pi)
HurdleNegativeBinomial(mu, theta, pi)

Arguments

`mu`	Location parameter of the negative binomial component of the distribution. Can be any positive number.
`theta`	Overdispersion parameter of the negative binomial component of the distribution. Can be any positive number.
`pi`	Zero-hurdle probability, can be any value in `⁠[0, 1]⁠`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a hurdle negative binomial random variable with parameters mu = $\mu$ and theta = $\theta$ .

Support: $\{0, 1, 2, 3, ...\}$

Mean:

$\mu \cdot \frac{\pi}{1 - F(0; \mu, \theta)}$

where $F(k; \mu)$ is the c.d.f. of the NegativeBinomial distribution.

Variance:

$m \cdot \left(1 + \frac{\mu}{\theta} + \mu - m \right)$

where $m$ is the mean above.

Probability mass function (p.m.f.): $P(X = 0) = 1 - \pi$ and for $k > 0$

$P(X = k) = \pi \cdot \frac{f(k; \mu, \theta)}{1 - F(0; \mu, \theta)}$

where $f(k; \mu, \theta)$ is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.): $P(X \le 0) = 1 - \pi$ and for $k > 0$

$P(X \le k) = 1 - \pi + \pi \cdot \frac{F(k; \mu, \theta) - F(0; \mu, \theta)}{1 - F(0; \mu, \theta)}$

Moment generating function (m.g.f.):

Omitted for now.

Value

A HurdleNegativeBinomial object.

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a hurdle Poisson distribution

Description

Hurdle Poisson distributions are frequently used to model counts with many zero observations.

Usage

HurdlePoisson(lambda, pi)
HurdlePoisson(lambda, pi)

Arguments

`lambda`	Parameter of the Poisson component of the distribution. Can be any positive number.
`pi`	Zero-hurdle probability, can be any value in `⁠[0, 1]⁠`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a hurdle Poisson random variable with parameter lambda = $\lambda$ .

Support: $\{0, 1, 2, 3, ...\}$

Mean:

$\lambda \cdot \frac{\pi}{1 - e^{-\lambda}}$

Variance: $m \cdot (\lambda + 1 - m)$ , where $m$ is the mean above.

Probability mass function (p.m.f.): $P(X = 0) = 1 - \pi$ and for $k > 0$

$P(X = k) = \pi \cdot \frac{f(k; \lambda)}{1 - f(0; \lambda)}$

where $f(k; \lambda)$ is the p.m.f. of the Poisson distribution.

Cumulative distribution function (c.d.f.): $P(X \le 0) = 1 - \pi$ and for $k > 0$

$P(X \le k) = 1 - \pi + \pi \cdot \frac{F(k; \lambda) - F(0; \lambda)}{1 - F(0; \lambda)}$

where $F(k; \lambda)$ is the c.d.f. of the Poisson distribution.

Moment generating function (m.g.f.):

Omitted for now.

Value

A HurdlePoisson object.

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a HyperGeometric distribution

Description

To understand the HyperGeometric distribution, consider a set of $r$ objects, of which $m$ are of the type I and $n$ are of the type II. A sample with size $k$ ( $k<r$ ) with no replacement is randomly chosen. The number of observed type I elements observed in this sample is set to be our random variable $X$ . For example, consider that in a set of 20 car parts, there are 4 that are defective (type I). If we take a sample of size 5 from those car parts, the probability of finding 2 that are defective will be given by the HyperGeometric distribution (needs double checking).

Usage

HyperGeometric(m, n, k)
HyperGeometric(m, n, k)

Arguments

`m`	The number of type I elements available.
`n`	The number of type II elements available.
`k`	The size of the sample taken.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a HyperGeometric random variable with success probability p = $p = m/(m+n)$ .

Support: $x \in { \{\max{(0, k-n)}, \dots, \min{(k,m)}}\}$

Mean: $\frac{km}{n+m} = kp$

Variance: $\frac{km(n)(n+m-k)}{(n+m)^2 (n+m-1)} = kp(1-p)(1 - \frac{k-1}{m+n-1})$

Probability mass function (p.m.f):

$P(X = x) = \frac{{m \choose x}{n \choose k-x}}{{m+n \choose k}}$

Cumulative distribution function (c.d.f):

$P(X \le k) \approx \Phi\Big(\frac{x - kp}{\sqrt{kp(1-p)}}\Big)$

Moment generating function (m.g.f):

Not useful.

Value

A HyperGeometric object.

Examples


set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine whether a distribution is discrete or continuous

Description

Generic functions for determining whether a certain probability distribution is discrete or continuous, respectively.

Usage

is_discrete(d, ...)

is_continuous(d, ...)
is_discrete(d, ...)

is_continuous(d, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Details

The generic function is_discrete is intended to return TRUE for every distribution whose entire support is discrete and FALSE otherwise. Analogously, is_continuous is intended to return TRUE for every distribution whose entire support is continuous and FALSE otherwise. For mixed discrete-continuous distributions both methods should return FALSE.

Methods for both generics are provided for all distribution classes set up in this package.

Value

A logical vector indicating whether the distribution(s) in d is/are discrete or continuous, respectively.

Examples

X <- Normal()
is_discrete(X)
is_continuous(X)
Y <- Binomial(size = 10, p = c(0.2, 0.5, 0.8))
is_discrete(Y)
is_continuous(Y)
X <- Normal()
is_discrete(X)
is_continuous(X)
Y <- Binomial(size = 10, p = c(0.2, 0.5, 0.8))
is_discrete(Y)
is_continuous(Y)

Is an object a distribution?

Description

is_distribution tests if x inherits from "distribution".

Usage

is_distribution(x)
is_distribution(x)

Arguments

`x`	An object to test.

Examples


Z <- Normal()

is_distribution(Z)
is_distribution(1L)
Z <- Normal()

is_distribution(Z)
is_distribution(1L)

Compute the (log-)likelihood of a probability distribution given data

Description

Functions for computing the (log-)likelihood based on a distribution object and observed data. The log-likelihood is computed as the sum of log-density contributions and the likelihood by taking the exponential thereof.

Usage

log_likelihood(d, x, ...)

likelihood(d, x, ...)
log_likelihood(d, x, ...)

likelihood(d, x, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`x`	A vector of data to compute the likelihood.
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Numeric value of the (log-)likelihood.

Examples

## distribution object
X <- Normal()
## sum of log_pdf() contributions
log_likelihood(X, c(-1, 0, 0, 0, 3))
## exp of log_likelihood()
likelihood(X, c(-1, 0, 0, 0, 3))
## distribution object
X <- Normal()
## sum of log_pdf() contributions
log_likelihood(X, c(-1, 0, 0, 0, 3))
## exp of log_likelihood()
likelihood(X, c(-1, 0, 0, 0, 3))

Create a Logistic distribution

Description

A continuous distribution on the real line. For binary outcomes the model given by $P(Y = 1 | X) = F(X \beta)$ where $F$ is the Logistic cdf() is called logistic regression.

Usage

Logistic(location = 0, scale = 1)
Logistic(location = 0, scale = 1)

Arguments

`location`	The location parameter for the distribution. For Logistic distributions, the location parameter is the mean, median and also mode. Defaults to zero.
`scale`	The scale parameter for the distribution. Defaults to one.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Logistic random variable with location = $\mu$ and scale = $s$ .

Support: $R$ , the set of all real numbers

Mean: $\mu$

Variance: $s^2 \pi^2 / 3$

Probability density function (p.d.f):

$f(x) = \frac{e^{-(\frac{x - \mu}{s})}}{s [1 + \exp(-(\frac{x - \mu}{s})) ]^2}$

Cumulative distribution function (c.d.f):

$F(t) = \frac{1}{1 + e^{-(\frac{t - \mu}{s})}}$

Moment generating function (m.g.f):

$E(e^{tX}) = e^{\mu t} \beta(1 - st, 1 + st)$

where $\beta(x, y)$ is the Beta function.

Value

A Logistic object.

Examples


set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a LogNormal distribution

Description

A random variable created by exponentiating a Normal() distribution. Taking the log of LogNormal data returns in Normal() data.

Usage

LogNormal(log_mu = 0, log_sigma = 1)
LogNormal(log_mu = 0, log_sigma = 1)

Arguments

`log_mu`	The location parameter, written $\mu$ in textbooks. Can be any real number. Defaults to `0`.
`log_sigma`	The scale parameter, written $\sigma$ in textbooks. Can be any positive real number. Defaults to `1`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a LogNormal random variable with success probability p = $p$ .

Support: $R^+$

Mean: $\exp(\mu + \sigma^2/2)$

Variance: $[\exp(\sigma^2)-1]\exp(2\mu+\sigma^2)$

Probability density function (p.d.f):

$f(x) = \frac{1}{x \sigma \sqrt{2 \pi}} \exp \left(-\frac{(\log x - \mu)^2}{2 \sigma^2} \right)$

Cumulative distribution function (c.d.f):

$F(x) = \frac{1}{2} + \frac{1}{2\sqrt{pi}}\int_{-x}^x e^{-t^2} dt$

Moment generating function (m.g.f): Undefined.

Value

A LogNormal object.

Examples


set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a Multinomial distribution

Description

The multinomial distribution is a generalization of the binomial distribution to multiple categories. It is perhaps easiest to think that we first extend a Bernoulli() distribution to include more than two categories, resulting in a Categorical() distribution. We then extend repeat the Categorical experiment several ( $n$ ) times.

Usage

Multinomial(size, p)
Multinomial(size, p)

Arguments

`size`	The number of trials. Must be an integer greater than or equal to one. When `size = 1L`, the Multinomial distribution reduces to the categorical distribution (also called the discrete uniform). Often called `n` in textbooks.
`p`	A vector of success probabilities for each trial. `p` can take on any positive value, and the vector is normalized internally.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X = (X_1, ..., X_k)$ be a Multinomial random variable with success probability p = $p$ . Note that $p$ is vector with $k$ elements that sum to one. Assume that we repeat the Categorical experiment size = $n$ times.

Support: Each $X_i$ is in ${0, 1, 2, ..., n}$ .

Mean: The mean of $X_i$ is $n p_i$ .

Variance: The variance of $X_i$ is $n p_i (1 - p_i)$ . For $i \neq j$ , the covariance of $X_i$ and $X_j$ is $-n p_i p_j$ .

Probability mass function (p.m.f):

$P(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot ... \cdot p_k^{x_k}$

Cumulative distribution function (c.d.f):

Omitted for multivariate random variables for the time being.

Moment generating function (m.g.f):

$E(e^{tX}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n$

Value

A Multinomial object.

Examples


set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)
set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)

Create a negative binomial distribution

Description

A generalization of the geometric distribution. It is the number of failures in a sequence of i.i.d. Bernoulli trials before a specified target number ( $r$ ) of successes occurs.

Usage

NegativeBinomial(size, p = 0.5, mu = size)
NegativeBinomial(size, p = 0.5, mu = size)

Arguments

`size`	The target number of successes (greater than $0$ ) until the experiment is stopped. Denoted $r$ below.
`p`	The success probability for a given trial. `p` can be any value in `⁠[0, 1]⁠`, and defaults to `0.5`.
`mu`	Alternative parameterization via the non-negative mean of the distribution (instead of the probability `p`), defaults to `size`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a negative binomial random variable with success probability p = $p$ .

Support: $\{0, 1, 2, 3, ...\}$

Mean: $\frac{(1 - p) r}{p} = \mu$

Variance: $\frac{(1 - p) r}{p^2}$

Probability mass function (p.m.f.):

$f(k) = {k + r - 1 \choose k} \cdot p^r (1-p)^k$

Cumulative distribution function (c.d.f.):

Omitted for now.

Moment generating function (m.g.f.):

$\left(\frac{p}{1 - (1 -p) e^t}\right)^r, t < -\log (1-p)$

Alternative parameterization: Sometimes, especially when used in regression models, the negative binomial distribution is parameterized by its mean $\mu$ (as listed above) plus the size parameter $r$ . This implies a success probability of $p = r/(r + \mu)$ . This can also be seen as a generalization of the Poisson distribution where the assumption of equidispersion (i.e., variance equal to mean) is relaxed. The negative binomial distribution is overdispersed (i.e., variance greater than mean) and its variance can also be written as $\mu + 1/r \mu^2$ . The Poisson distribution is then obtained as $r$ goes to infinity. Note that in this view it is natural to also allow for non-integer $r$ parameters. The factorials in the equations above are then expressed in terms of the gamma function.

Value

A NegativeBinomial object.

Examples


set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)
set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Create a Normal distribution

Description

The Normal distribution is ubiquitous in statistics, partially because of the central limit theorem, which states that sums of i.i.d. random variables eventually become Normal. Linear transformations of Normal random variables result in new random variables that are also Normal. If you are taking an intro stats course, you'll likely use the Normal distribution for Z-tests and in simple linear regression. Under regularity conditions, maximum likelihood estimators are asymptotically Normal. The Normal distribution is also called the gaussian distribution.

Usage

Normal(mu = 0, sigma = 1)
Normal(mu = 0, sigma = 1)

Arguments

`mu`	The location parameter, written $\mu$ in textbooks, which is also the mean of the distribution. Can be any real number. Defaults to `0`.
`sigma`	The scale parameter, written $\sigma$ in textbooks, which is also the standard deviation of the distribution. Can be any positive number. Defaults to `1`. If you would like a Normal distribution with variance $\sigma^2$ , be sure to take the square root, as this is a common source of errors.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Normal random variable with mean mu = $\mu$ and standard deviation sigma = $\sigma$ .

Support: $R$ , the set of all real numbers

Mean: $\mu$

Variance: $\sigma^2$

Probability density function (p.d.f):

$f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2}$

Cumulative distribution function (c.d.f):

The cumulative distribution function has the form

$F(t) = \int_{-\infty}^t \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-(x - \mu)^2 / 2 \sigma^2} dx$

but this integral does not have a closed form solution and must be approximated numerically. The c.d.f. of a standard Normal is sometimes called the "error function". The notation $\Phi(t)$ also stands for the c.d.f. of a standard Normal evaluated at $t$ . Z-tables list the value of $\Phi(t)$ for various $t$ .

Moment generating function (m.g.f):

$E(e^{tX}) = e^{\mu t + \sigma^2 t^2 / 2}$

Value

A Normal object.

Examples


set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability density of a probability distribution

Description

Generic function for computing probability density function (PDF) contributions based on a distribution object and observed data.

Usage

pdf(d, x, drop = TRUE, ...)

log_pdf(d, x, ...)

pmf(d, x, ...)
pdf(d, x, drop = TRUE, ...)

log_pdf(d, x, ...)

pmf(d, x, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Details

The generic function pdf() computes the probability density, both for continuous and discrete distributions. pmf() (for the probability mass function) is an alias that just calls pdf() internally. For computing log-density contributions (e.g., to a log-likelihood) either pdf(..., log = TRUE) can be used or the generic function log_pdf().

Value

Probabilities corresponding to the vector x.

Examples

## distribution object
X <- Normal()
## probability density
pdf(X, c(1, 2, 3, 4, 5))
pmf(X, c(1, 2, 3, 4, 5))
## log-density
pdf(X, c(1, 2, 3, 4, 5), log = TRUE)
log_pdf(X, c(1, 2, 3, 4, 5))
## distribution object
X <- Normal()
## probability density
pdf(X, c(1, 2, 3, 4, 5))
pmf(X, c(1, 2, 3, 4, 5))
## log-density
pdf(X, c(1, 2, 3, 4, 5), log = TRUE)
log_pdf(X, c(1, 2, 3, 4, 5))

Evaluate the probability mass function of a Bernoulli distribution

Description

Evaluate the probability mass function of a Bernoulli distribution

Usage

## S3 method for class 'Bernoulli'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Bernoulli'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Bernoulli'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Bernoulli'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Bernoulli` object created by a call to `Bernoulli()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Beta distribution

Description

Evaluate the probability mass function of a Beta distribution

Usage

## S3 method for class 'Beta'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Beta'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Beta'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Beta'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Beta` object created by a call to `Beta()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dbeta`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Binomial distribution

Description

Evaluate the probability mass function of a Binomial distribution

Usage

## S3 method for class 'Binomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Binomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Binomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Binomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Binomial` object created by a call to `Binomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Categorical discrete distribution

Description

Evaluate the probability mass function of a Categorical discrete distribution

Usage

## S3 method for class 'Categorical'
pdf(d, x, ...)

## S3 method for class 'Categorical'
log_pdf(d, x, ...)
## S3 method for class 'Categorical'
pdf(d, x, ...)

## S3 method for class 'Categorical'
log_pdf(d, x, ...)

Arguments

`d`	A `Categorical` object created by a call to `Categorical()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector of probabilities, one for each element of x.

Examples


set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)
set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Evaluate the probability mass function of a Cauchy distribution

Description

Evaluate the probability mass function of a Cauchy distribution

Usage

## S3 method for class 'Cauchy'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Cauchy'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Cauchy'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Cauchy'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Cauchy` object created by a call to `Cauchy()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dcauchy`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a chi square distribution

Description

Evaluate the probability mass function of a chi square distribution

Usage

## S3 method for class 'ChiSquare'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ChiSquare'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ChiSquare'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ChiSquare'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ChiSquare` object created by a call to `ChiSquare()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dchisq`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of an Erlang distribution

Description

Evaluate the probability mass function of an Erlang distribution

Usage

## S3 method for class 'Erlang'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Erlang'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Erlang'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Erlang'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	An `Erlang` object created by a call to `Erlang()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgamma`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability density function of an Exponential distribution

Description

Evaluate the probability density function of an Exponential distribution

Usage

## S3 method for class 'Exponential'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Exponential'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Exponential'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Exponential'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	An `Exponential` object created by a call to `Exponential()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dexp`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of an F distribution

Description

Evaluate the probability mass function of an F distribution

Usage

## S3 method for class 'FisherF'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'FisherF'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'FisherF'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'FisherF'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `FisherF` object created by a call to `FisherF()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `df`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Frechet distribution

Description

Evaluate the probability mass function of a Frechet distribution

Usage

## S3 method for class 'Frechet'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Frechet'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Frechet'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Frechet'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Frechet` object created by a call to `Frechet()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Gamma distribution

Description

Evaluate the probability mass function of a Gamma distribution

Usage

## S3 method for class 'Gamma'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Gamma'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Gamma'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Gamma'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Gamma` object created by a call to `Gamma()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgamma`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Geometric distribution

Description

Please see the documentation of Geometric() for some properties of the Geometric distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Geometric'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Geometric'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Geometric'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Geometric'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Geometric` object created by a call to `Geometric()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgeom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a GEV distribution

Description

Evaluate the probability mass function of a GEV distribution

Usage

## S3 method for class 'GEV'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'GEV'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'GEV'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'GEV'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `GEV` object created by a call to `GEV()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a GP distribution

Description

Evaluate the probability mass function of a GP distribution

Usage

## S3 method for class 'GP'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'GP'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'GP'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'GP'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `GP` object created by a call to `GP()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgp`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Gumbel distribution

Description

Evaluate the probability mass function of a Gumbel distribution

Usage

## S3 method for class 'Gumbel'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Gumbel'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Gumbel'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Gumbel'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Gumbel` object created by a call to `Gumbel()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a hurdle negative binomial distribution

Description

Evaluate the probability mass function of a hurdle negative binomial distribution

Usage

## S3 method for class 'HurdleNegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HurdleNegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HurdleNegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HurdleNegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `HurdleNegativeBinomial` object created by a call to `HurdleNegativeBinomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dhnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a hurdle Poisson distribution

Description

Evaluate the probability mass function of a hurdle Poisson distribution

Usage

## S3 method for class 'HurdlePoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HurdlePoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HurdlePoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HurdlePoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `HurdlePoisson` object created by a call to `HurdlePoisson()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dhpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a HyperGeometric distribution

Description

Please see the documentation of HyperGeometric() for some properties of the HyperGeometric distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'HyperGeometric'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HyperGeometric'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HyperGeometric'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'HyperGeometric'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `HyperGeometric` object created by a call to `HyperGeometric()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dhyper`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a Logistic distribution

Description

Please see the documentation of Logistic() for some properties of the Logistic distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Logistic'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Logistic'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Logistic'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Logistic'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Logistic` object created by a call to `Logistic()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dlogis`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a LogNormal distribution

Description

Please see the documentation of LogNormal() for some properties of the LogNormal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'LogNormal'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'LogNormal'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'LogNormal'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'LogNormal'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `LogNormal` object created by a call to `LogNormal()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dlnorm`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a Multinomial distribution

Description

Please see the documentation of Multinomial() for some properties of the Multinomial distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Multinomial'
pdf(d, x, ...)

## S3 method for class 'Multinomial'
log_pdf(d, x, ...)
## S3 method for class 'Multinomial'
pdf(d, x, ...)

## S3 method for class 'Multinomial'
log_pdf(d, x, ...)

Arguments

`d`	A `Multinomial` object created by a call to `Multinomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector of probabilities, one for each element of x.

Examples


set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)
set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)

Evaluate the probability mass function of a NegativeBinomial distribution

Description

Evaluate the probability mass function of a NegativeBinomial distribution

Usage

## S3 method for class 'NegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'NegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'NegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'NegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `NegativeBinomial` object created by a call to `NegativeBinomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)
set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Evaluate the probability mass function of a Normal distribution

Description

Please see the documentation of Normal() for some properties of the Normal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Normal'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Normal'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Normal'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Normal'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Normal` object created by a call to `Normal()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dnorm`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a Poisson distribution

Description

Evaluate the probability mass function of a Poisson distribution

Usage

## S3 method for class 'Poisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Poisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Poisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Poisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Poisson` object created by a call to `Poisson()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Evaluate the probability mass function of a PoissonBinomial distribution

Description

Evaluate the probability mass function of a PoissonBinomial distribution

Usage

## S3 method for class 'PoissonBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, log = FALSE, verbose = TRUE, ...)

## S3 method for class 'PoissonBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'PoissonBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, log = FALSE, verbose = TRUE, ...)

## S3 method for class 'PoissonBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `PoissonBinomial` object created by a call to `PoissonBinomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`log`, `...`	Arguments to be passed to `dpbinom` or `pnorm`, respectively.
`verbose`	logical. Should a warning be issued if the normal approximation is applied because the PoissonBinomial package is not installed?

Value

Examples


set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])
set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Evaluate the probability mass function of an RevWeibull distribution

Description

Evaluate the probability mass function of an RevWeibull distribution

Usage

## S3 method for class 'RevWeibull'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'RevWeibull'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'RevWeibull'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'RevWeibull'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `RevWeibull` object created by a call to `RevWeibull()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a StudentsT distribution

Description

Please see the documentation of StudentsT() for some properties of the StudentsT distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'StudentsT'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'StudentsT'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'StudentsT'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'StudentsT'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `StudentsT` object created by a call to `StudentsT()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dt`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Evaluate the probability mass function of a continuous Uniform distribution

Description

Evaluate the probability mass function of a continuous Uniform distribution

Usage

## S3 method for class 'Uniform'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Uniform'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Uniform'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Uniform'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Uniform` object created by a call to `Uniform()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dunif`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Evaluate the probability mass function of a Weibull distribution

Description

Please see the documentation of Weibull() for some properties of the Weibull distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Weibull'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Weibull'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Weibull'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'Weibull'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `Weibull` object created by a call to `Weibull()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dweibull`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Evaluate the probability mass function of a zero-inflated negative binomial distribution

Description

Evaluate the probability mass function of a zero-inflated negative binomial distribution

Usage

## S3 method for class 'ZINegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZINegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZINegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZINegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZINegativeBinomial` object created by a call to `ZINegativeBinomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dzinbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a zero-inflated Poisson distribution

Description

Evaluate the probability mass function of a zero-inflated Poisson distribution

Usage

## S3 method for class 'ZIPoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZIPoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZIPoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZIPoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZIPoisson` object created by a call to `ZIPoisson()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dzipois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a zero-truncated negative binomial distribution

Description

Evaluate the probability mass function of a zero-truncated negative binomial distribution

Usage

## S3 method for class 'ZTNegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZTNegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZTNegativeBinomial'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZTNegativeBinomial'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZTNegativeBinomial` object created by a call to `ZTNegativeBinomial()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dztnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Evaluate the probability mass function of a zero-truncated Poisson distribution

Description

Evaluate the probability mass function of a zero-truncated Poisson distribution

Usage

## S3 method for class 'ZTPoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZTPoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZTPoisson'
pdf(d, x, drop = TRUE, elementwise = NULL, ...)

## S3 method for class 'ZTPoisson'
log_pdf(d, x, drop = TRUE, elementwise = NULL, ...)

Arguments

`d`	A `ZTPoisson` object created by a call to `ZTPoisson()`.
`x`	A vector of elements whose probabilities you would like to determine given the distribution `d`.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `d` be evaluated at all elements of `x` (`elementwise = FALSE`, yielding a matrix)? Or, if `d` and `x` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `dztpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Plot the CDF of a distribution

Description

A function to easily plot the CDF of a distribution using ggplot2. Requires ggplot2 to be loaded.

Usage

plot_cdf(d, limits = NULL, p = 0.001, plot_theme = NULL)
plot_cdf(d, limits = NULL, p = 0.001, plot_theme = NULL)

Arguments

`d`	A `distribution` object
`limits`	either `NULL` (default) or a vector of length 2 that specifies the range of the x-axis
`p`	If `limits` is `NULL`, the range of the x-axis will be the support of `d` if this is a bounded interval, or `quantile(d, p)` and `quantile(d, 1 - p)` if lower and/or upper limits of the support is `-Inf`/`Inf`. Defaults to 0.001.
`plot_theme`	specify theme of resulting plot using `ggplot2`. Default is `theme_minimal`

Examples


N1 <- Normal()
plot_cdf(N1)

N2 <- Normal(0, c(1, 2))
plot_cdf(N2)

B1 <- Binomial(10, 0.2)
plot_cdf(B1)

B2 <- Binomial(10, c(0.2, 0.5))
plot_cdf(B2)
N1 <- Normal()
plot_cdf(N1)

N2 <- Normal(0, c(1, 2))
plot_cdf(N2)

B1 <- Binomial(10, 0.2)
plot_cdf(B1)

B2 <- Binomial(10, c(0.2, 0.5))
plot_cdf(B2)

Plot the PDF of a distribution

Description

A function to easily plot the PDF of a distribution using ggplot2. Requires ggplot2 to be loaded.

Usage

plot_pdf(d, limits = NULL, p = 0.001, plot_theme = NULL)
plot_pdf(d, limits = NULL, p = 0.001, plot_theme = NULL)

Arguments

`d`	A `distribution` object
`limits`	either `NULL` (default) or a vector of length 2 that specifies the range of the x-axis
`p`	If `limits` is `NULL`, the range of the x-axis will be the support of `d` if this is a bounded interval, or `quantile(d, p)` and `quantile(d, 1 - p)` if lower and/or upper limits of the support is `-Inf`/`Inf`. Defaults to 0.001.
`plot_theme`	specify theme of resulting plot using `ggplot2`. Default is `theme_minimal`

Examples


N1 <- Normal()
plot_pdf(N1)

N2 <- Normal(0, c(1, 2))
plot_pdf(N2)

B1 <- Binomial(10, 0.2)
plot_pdf(B1)

B2 <- Binomial(10, c(0.2, 0.5))
plot_pdf(B2)
N1 <- Normal()
plot_pdf(N1)

N2 <- Normal(0, c(1, 2))
plot_pdf(N2)

B1 <- Binomial(10, 0.2)
plot_pdf(B1)

B2 <- Binomial(10, c(0.2, 0.5))
plot_pdf(B2)

Plot the p.m.f, p.d.f or c.d.f. of a univariate distribution

Description

Plot method for an object inheriting from class "distribution". By default the probability density function (p.d.f.), for a continuous variable, or the probability mass function (p.m.f.), for a discrete variable, is plotted. The cumulative distribution function (c.d.f.) will be plotted if cdf = TRUE. Multiple functions are included in the plot if any of the parameter vectors in x has length greater than 1. See the argument all.

Usage

## S3 method for class 'distribution'
plot(
  x,
  cdf = FALSE,
  p = c(0.1, 99.9),
  len = 1000,
  all = FALSE,
  legend_args = list(),
  ...
)
## S3 method for class 'distribution'
plot(
  x,
  cdf = FALSE,
  p = c(0.1, 99.9),
  len = 1000,
  all = FALSE,
  legend_args = list(),
  ...
)

Arguments

`x`	an object of class `c("name", "distribution")`, where `"name"` is the name of the distribution.
`cdf`	A logical scalar. If `cdf = TRUE` then the cumulative distribution function (c.d.f.) is plotted. Otherwise, the probability density function (p.d.f.), for a continuous variable, or the probability mass function (p.m.f.), for a discrete variable, is plotted.
`p`	A numeric vector. If `xlim` is not passed in `...` then `p` is the fallback option for setting the range of values over which the p.m.f, p.d.f. or c.d.f is plotted. See Details.
`len`	An integer scalar. If `x` is a continuous distribution object then `len` is the number of values at which the p.d.f or c.d.f. is evaluated to produce the plot. The larger `len` is the smoother is the curve.
`all`	A logical scalar. If `all = TRUE` then a separate distribution is plotted for all the combinations of parameter values present in the parameter vectors present in `x`. These combinations are generated using `expand.grid`. If `all = FALSE` then the number of distributions plotted is equal to the maximum of the lengths of these parameter vectors, with shorter vectors recycled to this length if necessary using `rep_len`.
`legend_args`	A list of arguments to be passed to `legend`. In particular, the argument `x` (perhaps in conjunction with `legend_args$y`) can be used to set the position of the legend. If `legend_args$x` is not supplied then `"bottomright"` is used if `cdf = TRUE` and `"topright"` if `cdf = FALSE`.
`...`	Further arguments to be passed to `plot`, `plot.ecdf` and `lines`, such as `xlim, ylim, xlab, ylab, main, lwd, lty, col, pch`.

Details

If xlim is passed in ... then this determines the range of values of the variable to be plotted on the horizontal axis. If x is a discrete distribution object then the values for which the p.m.f. or c.d.f. is plotted is the smallest set of consecutive integers that contains both components of xlim. Otherwise, xlim is used directly.

If xlim is not passed in ... then the range of values spans the support of the distribution, with the following proviso: if the lower (upper) endpoint of the distribution is -Inf (Inf) then the lower (upper) limit of the plotting range is set to the p[1]\

If the name of x is a single upper case letter then that name is used to labels the axes of the plot. Otherwise, x and P(X = x) or f(x) are used.

A legend is included only if at least one of the parameter vectors in x has length greater than 1.

Plots of c.d.f.s are produced using calls to approxfun and plot.ecdf.

Value

An object with the same class as x, in which the parameter vectors have been expanded to contain a parameter combination for each function plotted.

Examples

B <- Binomial(20, 0.7)
plot(B)
plot(B, cdf = TRUE)

B2 <- Binomial(20, c(0.1, 0.5, 0.9))
plot(B2, legend_args = list(x = "top"))
x <- plot(B2, cdf = TRUE)
x$size
x$p

X <- Poisson(2)
plot(X)
plot(X, cdf = TRUE)

G <- Gamma(c(1, 3), 1:2)
plot(G)
plot(G, all = TRUE)
plot(G, cdf = TRUE)

C <- Cauchy()
plot(C, p = c(1, 99), len = 10000)
plot(C, cdf = TRUE, p = c(1, 99))
B <- Binomial(20, 0.7)
plot(B)
plot(B, cdf = TRUE)

B2 <- Binomial(20, c(0.1, 0.5, 0.9))
plot(B2, legend_args = list(x = "top"))
x <- plot(B2, cdf = TRUE)
x$size
x$p

X <- Poisson(2)
plot(X)
plot(X, cdf = TRUE)

G <- Gamma(c(1, 3), 1:2)
plot(G)
plot(G, all = TRUE)
plot(G, cdf = TRUE)

C <- Cauchy()
plot(C, p = c(1, 99), len = 10000)
plot(C, cdf = TRUE, p = c(1, 99))

Create a Poisson distribution

Description

Poisson distributions are frequently used to model counts.

Usage

Poisson(lambda)
Poisson(lambda)

Arguments

lambda

The shape parameter, which is also the mean and the variance of the distribution. Can be any positive number.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a Poisson random variable with parameter lambda = $\lambda$ .

Support: $\{0, 1, 2, 3, ...\}$

Mean: $\lambda$

Variance: $\lambda$

Probability mass function (p.m.f):

$P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}$

Cumulative distribution function (c.d.f):

$P(X \le k) = e^{-\lambda} \sum_{i = 0}^{\lfloor k \rfloor} \frac{\lambda^i}{i!}$

Moment generating function (m.g.f):

$E(e^{tX}) = e^{\lambda (e^t - 1)}$

Value

A Poisson object.

Examples


set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Create a Poisson binomial distribution

Description

The Poisson binomial distribution is a generalization of the Binomial distribution. It is also a sum of $n$ independent Bernoulli experiments. However, the success probabilities can vary between the experiments so that they are not identically distributed.

Usage

PoissonBinomial(...)
PoissonBinomial(...)

Arguments

...

An arbitrary number of numeric vectors or matrices of success probabilities in ⁠[0, 1]⁠ (with matching number of rows).

Details

The Poisson binomial distribution comes up when you consider the number of successes in independent binomial experiments (coin flips) with potentially varying success probabilities.

The PoissonBinomial distribution class in distributions3 is mostly based on the PoissonBinomial package, providing fast Rcpp implementations of efficient algorithms. Hence, it is recommended to install the PoissonBinomial package when working with this distribution. However, as a fallback for when the PoissonBinomial package is not installed the methods for the PoissonBinomial distribution employ a normal approximation.

We recommend reading the following documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a Poisson binomial random variable with success probabilities $p_1$ to $p_n$ .

Support: $\{0, 1, 2, ..., n\}$

Mean: $p_1 + \dots + p_n$

Variance: $p_1 \cdot (1 - p_1) + \dots + p_1 \cdot (1 - p_1)$

Probability mass function (p.m.f):

$P(X = k) = \sum_A \prod_{i \in A} p_i \prod_{j \in A^C} (1 - p_j)$

where the sum is taken over all sets $A$ with $k$ elements from $\{0, 1, 2, ..., n\}$ . $A^C$ is the complement of $A$ .

Cumulative distribution function (c.d.f):

$P(X \le k) = \sum_{i=0}^{\lfloor k \rfloor} P(X = i)$

Moment generating function (m.g.f):

$E(e^{tX}) = \prod_{i = 1}^n (1 - p_i + p_i e^t)$

Value

A PoissonBinomial object.

Examples


set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])
set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Extracting fitted or predicted probability distributions from models

Description

Generic function with methods for various model classes for extracting fitted (in-sample) or predicted (out-of-sample) probability distributions3 objects.

Usage

prodist(object, ...)

## S3 method for class 'lm'
prodist(object, ..., sigma = "ML")

## S3 method for class 'glm'
prodist(object, ..., dispersion = NULL)

## S3 method for class 'distribution'
prodist(object, ...)
prodist(object, ...)

## S3 method for class 'lm'
prodist(object, ..., sigma = "ML")

## S3 method for class 'glm'
prodist(object, ..., dispersion = NULL)

## S3 method for class 'distribution'
prodist(object, ...)

Arguments

`object`	A model object.
`...`	Arguments passed on to methods, typically for calling the underlying `predict` methods, e.g., `newdata` for `lm` or `glm` objects or `n.ahead` for `arima` objects.
`sigma`	character or numeric or `NULL`. Specification of the standard deviation `sigma` to be used for the `Normal` distribution in the `lm` method. The default `"ML"` (or equivalently `"MLE"` or `NULL`) uses the maximum likelihood estimate based on the residual sum of squares divided by the number of observations, n. Alternatively, `sigma = "OLS"` uses the least-squares estimate (divided by the residual degrees of freedom, n - k). Finally, a concrete numeric value can also be specified in `sigma`.
`dispersion`	character or numeric or `NULL`. Specification of the dispersion parameter in the `glm` method. The default `NULL` (or equivalently `"deviance"`) is to use the `deviance` divided by the number of observations, n. Alternatively, `dispersion = "Chisquared"` uses the Chi-squared statistic divided by the residual degrees of freedom, n - k. Finally, a concrete numeric value can also be specified in `dispersion`.

Details

To facilitate making probabilistic forecasts based on regression and time series model objects, the function prodist extracts fitted or predicted probability distribution objects. Currently, methods are provided for objects fitted by lm, glm, and arima in base R as well as glm.nb from the MASS package and hurdle/zeroinfl/zerotrunc from the pscl or countreg packages.

All methods essentially proceed in two steps: First, the standard predict method for these model objects is used to compute fitted (in-sample, default) or predicted (out-of-sample) distribution parameters. Typically, this includes the mean plus further parameters describing scale, dispersion, shape, etc.). Second, the distributions objects are set up using the generator functions from distributions3.

Note that these probability distributions only reflect the random variation in the dependent variable based on the model employed (and its associated distributional assumpation for the dependent variable). This does not capture the uncertainty in the parameter estimates.

For both linear regression models and generalized linear models, estimated by lm and glm respectively, there is some ambiguity as to which estimate for the dispersion parameter of the model is to be used. While the logLik methods use the maximum-likelihood (ML) estimate implicitly, the summary methods report an estimate that is standardized with the residual degrees of freedom, n - k (rather than the number of observations, n). The prodist methods for these objects follow the logLik method by default but the summary behavior can be mimicked by setting the sigma or dispersion arguments accordingly.

Value

An object inheriting from distribution.

Examples

## Model: Linear regression
## Fit: lm
## Data: 1920s cars data
data("cars", package = "datasets")

## Stopping distance (ft) explained by speed (mph)
reg <- lm(dist ~ speed, data = cars)

## Extract fitted normal distributions (in-sample, with constant variance)
pd <- prodist(reg)
head(pd)

## Extract log-likelihood from model object
logLik(reg)

## Replicate log-likelihood via distributions object
sum(log_pdf(pd, cars$dist))
log_likelihood(pd, cars$dist)

## Compute corresponding medians and 90% interval
qd <- quantile(pd, c(0.05, 0.5, 0.95))
head(qd)

## Visualize observations with predicted quantiles
plot(dist ~ speed, data = cars)
matplot(cars$speed, qd, add = TRUE, type = "l", col = 2, lty = 1)

## Sigma estimated by maximum-likelihood estimate (default, used in logLik)
## vs. least-squares estimate (used in summary)
nd <- data.frame(speed = 50)
prodist(reg, newdata = nd, sigma = "ML")
prodist(reg, newdata = nd, sigma = "OLS")
summary(reg)$sigma


## Model: Poisson generalized linear model
## Fit: glm
## Data: FIFA 2018 World Cup data
data("FIFA2018", package = "distributions3")

## Number of goals per team explained by ability differences
poisreg <- glm(goals ~ difference, data = FIFA2018, family = poisson)
summary(poisreg)
## Interpretation: When the ratio of abilities increases by 1 percent,
## the expected number of goals increases by around 0.4 percent

## Predict fitted Poisson distributions for teams with equal ability (out-of-sample)
nd <- data.frame(difference = 0)
prodist(poisreg, newdata = nd)

## Extract fitted Poisson distributions (in-sample)
pd <- prodist(poisreg)
head(pd)

## Extract log-likelihood from model object
logLik(poisreg)

## Replicate log-likelihood via distributions object
sum(log_pdf(pd, FIFA2018$goals))
log_likelihood(pd, FIFA2018$goals)


## Model: Autoregressive integrated moving average model
## Fit: arima
## Data: Quarterly approval ratings of U.S. presidents (1945-1974)
data("presidents", package = "datasets")

## ARMA(2,1) model
arma21 <- arima(presidents, order = c(2, 0, 1))

## Extract predicted normal distributions for next two years
p <- prodist(arma21, n.ahead = 8)
p

## Compute median (= mean) forecast along with 80% and 95% interval
quantile(p, c(0.5, 0.1, 0.9, 0.025, 0.975))

## Model: Linear regression
## Fit: lm
## Data: 1920s cars data
data("cars", package = "datasets")

## Stopping distance (ft) explained by speed (mph)
reg <- lm(dist ~ speed, data = cars)

## Extract fitted normal distributions (in-sample, with constant variance)
pd <- prodist(reg)
head(pd)

## Extract log-likelihood from model object
logLik(reg)

## Replicate log-likelihood via distributions object
sum(log_pdf(pd, cars$dist))
log_likelihood(pd, cars$dist)

## Compute corresponding medians and 90% interval
qd <- quantile(pd, c(0.05, 0.5, 0.95))
head(qd)

## Visualize observations with predicted quantiles
plot(dist ~ speed, data = cars)
matplot(cars$speed, qd, add = TRUE, type = "l", col = 2, lty = 1)

## Sigma estimated by maximum-likelihood estimate (default, used in logLik)
## vs. least-squares estimate (used in summary)
nd <- data.frame(speed = 50)
prodist(reg, newdata = nd, sigma = "ML")
prodist(reg, newdata = nd, sigma = "OLS")
summary(reg)$sigma


## Model: Poisson generalized linear model
## Fit: glm
## Data: FIFA 2018 World Cup data
data("FIFA2018", package = "distributions3")

## Number of goals per team explained by ability differences
poisreg <- glm(goals ~ difference, data = FIFA2018, family = poisson)
summary(poisreg)
## Interpretation: When the ratio of abilities increases by 1 percent,
## the expected number of goals increases by around 0.4 percent

## Predict fitted Poisson distributions for teams with equal ability (out-of-sample)
nd <- data.frame(difference = 0)
prodist(poisreg, newdata = nd)

## Extract fitted Poisson distributions (in-sample)
pd <- prodist(poisreg)
head(pd)

## Extract log-likelihood from model object
logLik(poisreg)

## Replicate log-likelihood via distributions object
sum(log_pdf(pd, FIFA2018$goals))
log_likelihood(pd, FIFA2018$goals)


## Model: Autoregressive integrated moving average model
## Fit: arima
## Data: Quarterly approval ratings of U.S. presidents (1945-1974)
data("presidents", package = "datasets")

## ARMA(2,1) model
arma21 <- arima(presidents, order = c(2, 0, 1))

## Extract predicted normal distributions for next two years
p <- prodist(arma21, n.ahead = 8)
p

## Compute median (= mean) forecast along with 80% and 95% interval
quantile(p, c(0.5, 0.1, 0.9, 0.025, 0.975))

Determine quantiles of a Bernoulli distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Bernoulli'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Bernoulli'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Bernoulli` object created by a call to `Bernoulli()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object, either a numeric vector of length probs (if drop = TRUE, default) or a matrix with length(probs) columns (if drop = FALSE). In case of a vectorized distribution object, a matrix with length(probs) columns containing all possible combinations.

Examples


set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a Beta distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Beta'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Beta'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Beta` object created by a call to `Beta()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qbeta`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a Binomial distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Binomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Binomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Binomial` object created by a call to `Binomial()`.
`probs`	A vector of probabilities.
`drop`	logical. Shoul the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of a Categorical discrete distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Categorical'
quantile(x, probs, ...)
## S3 method for class 'Categorical'
quantile(x, probs, ...)

Arguments

`x`	A `Categorical` object created by a call to `Categorical()`.
`probs`	A vector of probabilities.
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector of quantiles, one for each element of probs.

Examples


set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)
set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Determine quantiles of a Cauchy distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Cauchy'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Cauchy'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Cauchy` object created by a call to `Cauchy()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qcauchy`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of a chi square distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'ChiSquare'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ChiSquare'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `ChiSquare` object created by a call to `ChiSquare()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qchisq`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of an Erlang distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Erlang'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Erlang'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	An `Erlang` object created by a call to `Erlang()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgamma`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of an Exponential distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Exponential'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Exponential'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	An `Exponential` object created by a call to `Exponential()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qexp`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of an F distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'FisherF'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'FisherF'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `FisherF` object created by a call to `FisherF()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qf`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of a Frechet distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Frechet'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Frechet'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Frechet` object created by a call to `Frechet()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a Gamma distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Gamma'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Gamma'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Gamma` object created by a call to `Gamma()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgamma`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of a Geometric distribution

Description

Determine quantiles of a Geometric distribution

Usage

## S3 method for class 'Geometric'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Geometric'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Geometric` object created by a call to `Geometric()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgeom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine quantiles of a GEV distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'GEV'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'GEV'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `GEV` object created by a call to `GEV()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a GP distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'GP'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'GP'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `GP` object created by a call to `GP()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgp`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a Gumbel distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Gumbel'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Gumbel'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Gumbel` object created by a call to `Gumbel()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a hurdle negative binomial distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'HurdleNegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HurdleNegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `HurdleNegativeBinomial` object created by a call to `HurdleNegativeBinomial()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qhnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Determine quantiles of a hurdle Poisson distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'HurdlePoisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HurdlePoisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `HurdlePoisson` object created by a call to `HurdlePoisson()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qhpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Determine quantiles of a HyperGeometric distribution

Description

Determine quantiles of a HyperGeometric distribution

Usage

## S3 method for class 'HyperGeometric'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'HyperGeometric'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `HyperGeometric` object created by a call to `HyperGeometric()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qhyper`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine quantiles of a Logistic distribution

Description

Determine quantiles of a Logistic distribution

Usage

## S3 method for class 'Logistic'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Logistic'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Logistic` object created by a call to `Logistic()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qlogis`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine quantiles of a LogNormal distribution

Description

Determine quantiles of a LogNormal distribution

Usage

## S3 method for class 'LogNormal'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'LogNormal'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `LogNormal` object created by a call to `LogNormal()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qlnorm`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine quantiles of a NegativeBinomial distribution

Description

Determine quantiles of a NegativeBinomial distribution

Usage

## S3 method for class 'NegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'NegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `NegativeBinomial` object created by a call to `NegativeBinomial()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)
set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Determine quantiles of a Normal distribution

Description

Please see the documentation of Normal() for some properties of the Normal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals. quantile()

Usage

## S3 method for class 'Normal'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Normal'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Normal` object created by a call to `Normal()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qnorm`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Details

This function returns the same values that you get from a Z-table. Note quantile() is the inverse of cdf(). Please see the documentation of Normal() for some properties of the Normal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Value

Examples


set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of a Poisson distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Poisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Poisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Poisson` object created by a call to `Poisson()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Determine quantiles of a PoissonBinomial distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'PoissonBinomial'
quantile(
  x,
  probs,
  drop = TRUE,
  elementwise = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  verbose = TRUE,
  ...
)
## S3 method for class 'PoissonBinomial'
quantile(
  x,
  probs,
  drop = TRUE,
  elementwise = NULL,
  lower.tail = TRUE,
  log.p = FALSE,
  verbose = TRUE,
  ...
)

Arguments

`x`	A `PoissonBinomial` object created by a call to `PoissonBinomial()`.
`probs`	A vector of probabilities.
`drop`	logical. Shoul the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`lower.tail`, `log.p`, `...`	Arguments to be passed to `qpbinom` or `qnorm`, respectively.
`verbose`	logical. Should a warning be issued if the normal approximation is applied because the PoissonBinomial package is not installed?

Value

Examples


set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])
set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Determine quantiles of a RevWeibull distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'RevWeibull'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'RevWeibull'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `RevWeibull` object created by a call to `RevWeibull()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qgev`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a StudentsT distribution

Description

Please see the documentation of StudentsT() for some properties of the StudentsT distribution, as well as extensive examples showing to how calculate p-values and confidence intervals. quantile()

Usage

## S3 method for class 'StudentsT'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'StudentsT'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `StudentsT` object created by a call to `StudentsT()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qt`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Details

This function returns the same values that you get from a Z-table. Note quantile() is the inverse of cdf(). Please see the documentation of StudentsT() for some properties of the StudentsT distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Value

Examples


set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Determine quantiles of a Tukey distribution

Description

Determine quantiles of a Tukey distribution

Usage

## S3 method for class 'Tukey'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Tukey'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A vector of elements whose cumulative probabilities you would like to determine given the distribution `d`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qtukey`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)

Determine quantiles of a continuous Uniform distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'Uniform'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Uniform'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Uniform` object created by a call to `Uniform()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qunif`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Determine quantiles of a Weibull distribution

Description

Determine quantiles of a Weibull distribution

Usage

## S3 method for class 'Weibull'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'Weibull'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `Weibull` object created by a call to `Weibull()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qweibull`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Determine quantiles of a zero-inflated negative binomial distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'ZINegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZINegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `ZINegativeBinomial` object created by a call to `ZINegativeBinomial()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qzinbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Determine quantiles of a zero-inflated Poisson distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'ZIPoisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZIPoisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `ZIPoisson` object created by a call to `ZIPoisson()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qzipois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Determine quantiles of a zero-truncated negative binomial distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'ZTNegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZTNegativeBinomial'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `ZTNegativeBinomial` object created by a call to `ZTNegativeBinomial()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qztnbinom`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Determine quantiles of a zero-truncated Poisson distribution

Description

quantile() is the inverse of cdf().

Usage

## S3 method for class 'ZTPoisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)
## S3 method for class 'ZTPoisson'
quantile(x, probs, drop = TRUE, elementwise = NULL, ...)

Arguments

`x`	A `ZTPoisson` object created by a call to `ZTPoisson()`.
`probs`	A vector of probabilities.
`drop`	logical. Should the result be simplified to a vector if possible?
`elementwise`	logical. Should each distribution in `x` be evaluated at all elements of `probs` (`elementwise = FALSE`, yielding a matrix)? Or, if `x` and `probs` have the same length, should the evaluation be done element by element (`elementwise = TRUE`, yielding a vector)? The default of `NULL` means that `elementwise = TRUE` is used if the lengths match and otherwise `elementwise = FALSE` is used.
`...`	Arguments to be passed to `qztpois`. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Draw a random sample from a probability distribution

Description

Generic function for drawing random samples from distribution objects.

Usage

random(x, n = 1L, drop = TRUE, ...)

## S3 method for class 'distribution'
simulate(object, nsim = 1L, seed = NULL, ...)
random(x, n = 1L, drop = TRUE, ...)

## S3 method for class 'distribution'
simulate(object, nsim = 1L, seed = NULL, ...)

Arguments

`x`, `object`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`n`, `nsim`	The number of samples to draw. Should be a positive integer. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.
`seed`	An optional random seed that is to be set using `set.seed` prior to drawing the random sample. The previous random seed from the global environment (if any) is restored afterwards.

Details

random is a new generic for drawing random samples from the S3 distribution objects provided in this package, such as Normal or Binomial etc. The respective methods typically call the "r" function for the corresponding distribution functions provided in base R such as rnorm, rbinom etc.

In addition to the new random generic there is also a simulate method for distribution objects which simply calls the random method internally.

Value

Random samples drawn from the distriubtion x. The random methods typically return either a matrix or, if possible, a vector. The simulate method always returns a data frame (with an attribute "seed" containing the .Random.seed from before the simulation).

Examples

## distribution object
X <- Normal()
## 10 random samples
random(X, 10)
## distribution object
X <- Normal()
## 10 random samples
random(X, 10)

Draw a random sample from a Bernoulli distribution

Description

Draw a random sample from a Bernoulli distribution

Usage

## S3 method for class 'Bernoulli'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Bernoulli'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Bernoulli` object created by a call to `Bernoulli()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Bernoulli(0.7)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)
pdf(X, 1)
log_pdf(X, 1)
cdf(X, 0)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a Beta distribution

Description

Draw a random sample from a Beta distribution

Usage

## S3 method for class 'Beta'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Beta'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Beta` object created by a call to `Beta()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Values in ⁠[0, 1]⁠. In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Beta(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

mean(X)
variance(X)
skewness(X)
kurtosis(X)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a Binomial distribution

Description

Draw a random sample from a Binomial distribution

Usage

## S3 method for class 'Binomial'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Binomial'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Binomial` object created by a call to `Binomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Integers containing values between 0 and x$size. In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Binomial(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2L)
log_pdf(X, 2L)

cdf(X, 4L)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from a Categorical distribution

Description

Draw a random sample from a Categorical distribution

Usage

## S3 method for class 'Categorical'
random(x, n = 1L, ...)
## S3 method for class 'Categorical'
random(x, n = 1L, ...)

Arguments

`x`	A `Categorical` object created by a call to `Categorical()`.
`n`	The number of samples to draw. Defaults to `1L`.
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector containing values from outcomes of length n.

Examples


set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)
set.seed(27)

X <- Categorical(1:3, p = c(0.4, 0.1, 0.5))
X

Y <- Categorical(LETTERS[1:4])
Y

random(X, 10)
random(Y, 10)

pdf(X, 1)
log_pdf(X, 1)

cdf(X, 1)
quantile(X, 0.5)

# cdfs are only defined for numeric sample spaces. this errors!
# cdf(Y, "a")

# same for quantiles. this also errors!
# quantile(Y, 0.7)

Draw a random sample from a Cauchy distribution

Description

Draw a random sample from a Cauchy distribution

Usage

## S3 method for class 'Cauchy'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Cauchy'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Cauchy` object created by a call to `Cauchy()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Cauchy(10, 0.2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from a chi square distribution

Description

Draw a random sample from a chi square distribution

Usage

## S3 method for class 'ChiSquare'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'ChiSquare'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `ChiSquare` object created by a call to `ChiSquare()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- ChiSquare(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from an Erlang distribution

Description

Draw a random sample from an Erlang distribution

Usage

## S3 method for class 'Erlang'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Erlang'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	An `Erlang` object created by a call to `Erlang()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Erlang(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from an Exponential distribution

Description

Draw a random sample from an Exponential distribution

Usage

## S3 method for class 'Exponential'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Exponential'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	An `Exponential` object created by a call to `Exponential()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Exponential(5)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from an F distribution

Description

Draw a random sample from an F distribution

Usage

## S3 method for class 'FisherF'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'FisherF'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `FisherF` object created by a call to `FisherF()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- FisherF(5, 10, 0.2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from a Frechet distribution

Description

Draw a random sample from a Frechet distribution

Usage

## S3 method for class 'Frechet'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Frechet'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Frechet` object created by a call to `Frechet()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Frechet(0, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a Gamma distribution

Description

Draw a random sample from a Gamma distribution

Usage

## S3 method for class 'Gamma'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Gamma'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Gamma` object created by a call to `Gamma()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Gamma(5, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from a Geometric distribution

Description

Please see the documentation of Geometric() for some properties of the Geometric distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Geometric'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Geometric'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Geometric` object created by a call to `Geometric()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Geometric(0.3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Draw a random sample from a GEV distribution

Description

Draw a random sample from a GEV distribution

Usage

## S3 method for class 'GEV'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'GEV'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `GEV` object created by a call to `GEV()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GEV(1, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a GP distribution

Description

Draw a random sample from a GP distribution

Usage

## S3 method for class 'GP'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'GP'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `GP` object created by a call to `GP()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- GP(0, 2, 0.1)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a Gumbel distribution

Description

Draw a random sample from a Gumbel distribution

Usage

## S3 method for class 'Gumbel'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Gumbel'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Gumbel` object created by a call to `Gumbel()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a hurdle negative binomial distribution

Description

Draw a random sample from a hurdle negative binomial distribution

Usage

## S3 method for class 'HurdleNegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'HurdleNegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `HurdleNegativeBinomial` object created by a call to `HurdleNegativeBinomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples

## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle negative binomial distribution
X <- HurdleNegativeBinomial(mu = 2.5, theta = 1, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Draw a random sample from a hurdle Poisson distribution

Description

Draw a random sample from a hurdle Poisson distribution

Usage

## S3 method for class 'HurdlePoisson'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'HurdlePoisson'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `HurdlePoisson` object created by a call to `HurdlePoisson()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples

## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a hurdle Poisson distribution
X <- HurdlePoisson(lambda = 2.5, pi = 0.75)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Draw a random sample from a HyperGeometric distribution

Description

Please see the documentation of HyperGeometric() for some properties of the HyperGeometric distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'HyperGeometric'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'HyperGeometric'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `HyperGeometric` object created by a call to `HyperGeometric()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- HyperGeometric(4, 5, 8)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Draw a random sample from a Logistic distribution

Description

Draw a random sample from a Logistic distribution

Usage

## S3 method for class 'Logistic'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Logistic'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Logistic` object created by a call to `Logistic()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Logistic(2, 4)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Draw a random sample from a LogNormal distribution

Description

Draw a random sample from a LogNormal distribution

Usage

## S3 method for class 'LogNormal'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'LogNormal'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `LogNormal` object created by a call to `LogNormal()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- LogNormal(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Draw a random sample from a Multinomial distribution

Description

Draw a random sample from a Multinomial distribution

Usage

## S3 method for class 'Multinomial'
random(x, n = 1L, ...)
## S3 method for class 'Multinomial'
random(x, n = 1L, ...)

Arguments

`x`	A `Multinomial` object created by a call to `Multinomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

An integer vector of length n.

Examples


set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)
set.seed(27)

X <- Multinomial(size = 5, p = c(0.3, 0.4, 0.2, 0.1))
X

random(X, 10)

# pdf(X, 2)
# log_pdf(X, 2)

Draw a random sample from a negative binomial distribution

Description

Draw a random sample from a negative binomial distribution

Usage

## S3 method for class 'NegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'NegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `NegativeBinomial` object created by a call to `NegativeBinomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)
set.seed(27)

X <- NegativeBinomial(size = 5, p = 0.1)
X

random(X, 10)

pdf(X, 50)
log_pdf(X, 50)

cdf(X, 50)
quantile(X, 0.7)

## alternative parameterization of X
Y <- NegativeBinomial(mu = 45, size = 5)
Y
cdf(Y, 50)
quantile(Y, 0.7)

Draw a random sample from a Normal distribution

Description

Please see the documentation of Normal() for some properties of the Normal distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'Normal'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Normal'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Normal` object created by a call to `Normal()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Normal(5, 2)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided Z-test

# here the null hypothesis is H_0: mu = 3
# and we assume sigma = 2

# exactly the same as: Z <- Normal(0, 1)
Z <- Normal()

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the z-statistic
z_stat <- (mean(x) - 3) / (2 / sqrt(nx))
z_stat

# calculate the two-sided p-value
1 - cdf(Z, abs(z_stat)) + cdf(Z, -abs(z_stat))

# exactly equivalent to the above
2 * cdf(Z, -abs(z_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(Z, z_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(Z, z_stat)

### example: calculating a 88 percent Z CI for a mean

# same `x` as before, still assume `sigma = 2`

# lower-bound
mean(x) - quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# upper-bound
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

# also equivalent to
mean(x) + quantile(Z, 0.12 / 2) * 2 / sqrt(nx)
mean(x) + quantile(Z, 1 - 0.12 / 2) * 2 / sqrt(nx)

### generating random samples and plugging in ks.test()

set.seed(27)

# generate a random sample
ns <- random(Normal(3, 7), 26)

# test if sample is Normal(3, 7)
ks.test(ns, pnorm, mean = 3, sd = 7)

# test if sample is gamma(8, 3) using base R pgamma()
ks.test(ns, pgamma, shape = 8, rate = 3)

### MISC

# note that the cdf() and quantile() functions are inverses
cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from a Poisson distribution

Description

Draw a random sample from a Poisson distribution

Usage

## S3 method for class 'Poisson'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Poisson'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Poisson` object created by a call to `Poisson()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))
set.seed(27)

X <- Poisson(2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 7))

Draw a random sample from a PoissonBinomial distribution

Description

Draw a random sample from a PoissonBinomial distribution

Usage

## S3 method for class 'PoissonBinomial'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'PoissonBinomial'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `PoissonBinomial` object created by a call to `PoissonBinomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Examples


set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])
set.seed(27)

X <- PoissonBinomial(0.5, 0.3, 0.8)
X

mean(X)
variance(X)
skewness(X)
kurtosis(X)

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 2)
quantile(X, 0.8)

cdf(X, quantile(X, 0.8))
quantile(X, cdf(X, 2))

## equivalent definitions of four Poisson binomial distributions
## each summing up three Bernoulli probabilities
p <- cbind(
  p1 = c(0.1, 0.2, 0.1, 0.2),
  p2 = c(0.5, 0.5, 0.5, 0.5),
  p3 = c(0.8, 0.7, 0.9, 0.8))
PoissonBinomial(p)
PoissonBinomial(p[, 1], p[, 2], p[, 3])
PoissonBinomial(p[, 1:2], p[, 3])

Draw a random sample from an RevWeibull distribution

Description

Draw a random sample from an RevWeibull distribution

Usage

## S3 method for class 'RevWeibull'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'RevWeibull'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `RevWeibull` object created by a call to `RevWeibull()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a StudentsT distribution

Description

Please see the documentation of StudentsT() for some properties of the T distribution, as well as extensive examples showing to how calculate p-values and confidence intervals.

Usage

## S3 method for class 'StudentsT'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'StudentsT'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `StudentsT` object created by a call to `StudentsT()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Draw a random sample from a Tukey distribution

Description

Draw a random sample from a Tukey distribution

Usage

## S3 method for class 'Tukey'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Tukey'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Tukey` object created by a call to `Tukey()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)

Draw a random sample from a continuous Uniform distribution

Description

Draw a random sample from a continuous Uniform distribution

Usage

## S3 method for class 'Uniform'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Uniform'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Uniform` object created by a call to `Uniform()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

Values in ⁠[a, b]⁠. In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Draw a random sample from a Weibull distribution

Description

Draw a random sample from a Weibull distribution

Usage

## S3 method for class 'Weibull'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'Weibull'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `Weibull` object created by a call to `Weibull()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples


set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Draw a random sample from a zero-inflated negative binomial distribution

Description

Draw a random sample from a zero-inflated negative binomial distribution

Usage

## S3 method for class 'ZINegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'ZINegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `ZINegativeBinomial` object created by a call to `ZINegativeBinomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Draw a random sample from a zero-inflated Poisson distribution

Description

Draw a random sample from a zero-inflated Poisson distribution

Usage

## S3 method for class 'ZIPoisson'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'ZIPoisson'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `ZIPoisson` object created by a call to `ZIPoisson()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Draw a random sample from a zero-truncated negative binomial distribution

Description

Draw a random sample from a zero-truncated negative binomial distribution

Usage

## S3 method for class 'ZTNegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'ZTNegativeBinomial'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `ZTNegativeBinomial` object created by a call to `ZTNegativeBinomial()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Draw a random sample from a zero-truncated Poisson distribution

Description

Draw a random sample from a zero-truncated Poisson distribution

Usage

## S3 method for class 'ZTPoisson'
random(x, n = 1L, drop = TRUE, ...)
## S3 method for class 'ZTPoisson'
random(x, n = 1L, drop = TRUE, ...)

Arguments

`x`	A `ZTPoisson` object created by a call to `ZTPoisson()`.
`n`	The number of samples to draw. Defaults to `1L`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Unused. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

In case of a single distribution object or n = 1, either a numeric vector of length n (if drop = TRUE, default) or a matrix with n columns (if drop = FALSE).

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a reversed Weibull distribution

Description

The reversed (or negated) Weibull distribution is a special case of the ⁠\link{GEV}⁠ distribution, obtained when the GEV shape parameter $\xi$ is negative. It may be referred to as a type III extreme value distribution.

Usage

RevWeibull(location = 0, scale = 1, shape = 1)
RevWeibull(location = 0, scale = 1, shape = 1)

Arguments

`location`	The location (maximum) parameter $m$ . `location` can be any real number. Defaults to `0`.
`scale`	The scale parameter $s$ . `scale` can be any positive number. Defaults to `1`.
`shape`	The scale parameter $\alpha$ . `shape` can be any positive number. Defaults to `1`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a reversed Weibull random variable with location parameter location = $m$ , scale parameter scale = $s$ , and shape parameter shape = $\alpha$ . An RevWeibull( $m, s, \alpha$ ) distribution is equivalent to a ⁠\link{GEV}⁠( $m - s, s / \alpha, -1 / \alpha$ ) distribution.

If $X$ has an RevWeibull( $m, \lambda, k$ ) distribution then $m - X$ has a ⁠\link{Weibull}⁠( $k, \lambda$ ) distribution, that is, a Weibull distribution with shape parameter $k$ and scale parameter $\lambda$ .

Support: $(-\infty, m)$ .

Mean: $m + s\Gamma(1 + 1/\alpha)$ .

Median: $m + s(\ln 2)^{1/\alpha}$ .

Variance: $s^2 [\Gamma(1 + 2 / \alpha) - \Gamma(1 + 1 / \alpha)^2]$ .

Probability density function (p.d.f):

$f(x) = \alpha s ^ {-1} [-(x - m) / s] ^ {\alpha - 1}% \exp\{-[-(x - m) / s] ^ {\alpha} \}$

for $x < m$ . The p.d.f. is 0 for $x \geq m$ .

Cumulative distribution function (c.d.f):

$F(x) = \exp\{-[-(x - m) / s] ^ {\alpha} \}$

for $x < m$ . The c.d.f. is 1 for $x \geq m$ .

Value

A RevWeibull object.

Examples


set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- RevWeibull(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Simulate responses from fitted model objects

Description

Default method for simulating new responses from any model object with a prodist method (for extracting a probability distribution object).

Usage

## Default S3 method:
simulate(object, nsim = 1, seed = NULL, ...)
## Default S3 method:
simulate(object, nsim = 1, seed = NULL, ...)

Arguments

`object`	An object for which a `prodist` method is available.
`nsim`	The number of response vectors to simulate. Should be a positive integer. Defaults to 1.
`seed`	An optional random seed that is to be set using `set.seed` prior to drawing the random sample. The previous random seed from the global environment (if any) is restored afterwards.
`...`	Arguments passed to `simulate.distribution`.

Details

This default method simply combines two building blocks provided in this package: (1) prodist for extracting the probability distribution from a fitted model object, (2) simulate.distribution for simulating new observations from this distribution (internally calling random).

Thus, this enables simulation from any fitted model object that provides a prodist method. It waives the need to implement a dedicated simulate method for this model class.

Value

A data frame with an attribute "seed" containing the .Random.seed from before the simulation.

Examples

## Poisson GLM for FIFA 2018 goals
data("FIFA2018", package = "distributions3")
m <- glm(goals ~ difference, data = FIFA2018, family = poisson)

## simulate new goals via glm method
set.seed(0)
g_glm <- simulate(m, n = 3)

## alternatively use the new default method
set.seed(0)
g_default <- simulate.default(m, n = 3)

## same results
all.equal(g_glm, g_default, check.attributes = FALSE)
## Poisson GLM for FIFA 2018 goals
data("FIFA2018", package = "distributions3")
m <- glm(goals ~ difference, data = FIFA2018, family = poisson)

## simulate new goals via glm method
set.seed(0)
g_glm <- simulate(m, n = 3)

## alternatively use the new default method
set.seed(0)
g_default <- simulate.default(m, n = 3)

## same results
all.equal(g_glm, g_default, check.attributes = FALSE)

Fill out area under the curve for a plotted PDF

Description

Fill out area under the curve for a plotted PDF

Usage

stat_auc(
  mapping = NULL,
  data = NULL,
  geom = "auc",
  position = "identity",
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  from = -Inf,
  to = Inf,
  annotate = FALSE,
  digits = 3,
  ...
)

geom_auc(
  mapping = NULL,
  data = NULL,
  stat = "auc",
  position = "identity",
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  from = -Inf,
  to = Inf,
  annotate = FALSE,
  digits = 3,
  ...
)
stat_auc(
  mapping = NULL,
  data = NULL,
  geom = "auc",
  position = "identity",
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  from = -Inf,
  to = Inf,
  annotate = FALSE,
  digits = 3,
  ...
)

geom_auc(
  mapping = NULL,
  data = NULL,
  stat = "auc",
  position = "identity",
  na.rm = FALSE,
  show.legend = NA,
  inherit.aes = TRUE,
  from = -Inf,
  to = Inf,
  annotate = FALSE,
  digits = 3,
  ...
)

Arguments

`mapping`	Set of aesthetic mappings created by `aes()`. If specified and `inherit.aes = TRUE` (the default), it is combined with the default mapping at the top level of the plot. You must supply `mapping` if there is no plot mapping.
`data`	The data to be displayed in this layer. There are three options: If `NULL`, the default, the data is inherited from the plot data as specified in the call to `ggplot()`. A `data.frame`, or other object, will override the plot data. All objects will be fortified to produce a data frame. See `fortify()` for which variables will be created. A `function` will be called with a single argument, the plot data. The return value must be a `data.frame`, and will be used as the layer data. A `function` can be created from a `formula` (e.g. `~ head(.x, 10)`).
`geom`	The geometric object to use to display the data for this layer. When using a `⁠stat_*()⁠` function to construct a layer, the `geom` argument can be used to override the default coupling between stats and geoms. The `geom` argument accepts the following: A `Geom` ggproto subclass, for example `GeomPoint`. A string naming the geom. To give the geom as a string, strip the function name of the `geom_` prefix. For example, to use `geom_point()`, give the geom as `"point"`. For more information and other ways to specify the geom, see the layer geom documentation.
`position`	A position adjustment to use on the data for this layer. This can be used in various ways, including to prevent overplotting and improving the display. The `position` argument accepts the following: The result of calling a position function, such as `position_jitter()`. This method allows for passing extra arguments to the position. A string naming the position adjustment. To give the position as a string, strip the function name of the `position_` prefix. For example, to use `position_jitter()`, give the position as `"jitter"`. For more information and other ways to specify the position, see the layer position documentation.
`na.rm`	If `FALSE`, the default, missing values are removed with a warning. If `TRUE`, missing values are silently removed.
`show.legend`	logical. Should this layer be included in the legends? `NA`, the default, includes if any aesthetics are mapped. `FALSE` never includes, and `TRUE` always includes. It can also be a named logical vector to finely select the aesthetics to display.
`inherit.aes`	If `FALSE`, overrides the default aesthetics, rather than combining with them. This is most useful for helper functions that define both data and aesthetics and shouldn't inherit behaviour from the default plot specification, e.g. `borders()`.
`from`	Left end-point of interval
`to`	Right end-point of interval
`annotate`	Should P() be added in the upper left corner as an annotation? Works also with a colour character, e.g., "red".
`digits`	Number of digits shown in annotation
`...`	Other arguments passed on to `layer()`'s `params` argument. These arguments broadly fall into one of 4 categories below. Notably, further arguments to the `position` argument, or aesthetics that are required can not be passed through `...`. Unknown arguments that are not part of the 4 categories below are ignored. Static aesthetics that are not mapped to a scale, but are at a fixed value and apply to the layer as a whole. For example, `colour = "red"` or `linewidth = 3`. The geom's documentation has an Aesthetics section that lists the available options. The 'required' aesthetics cannot be passed on to the `params`. Please note that while passing unmapped aesthetics as vectors is technically possible, the order and required length is not guaranteed to be parallel to the input data. When constructing a layer using a `⁠stat_()⁠` function, the `...` argument can be used to pass on parameters to the `geom` part of the layer. An example of this is `stat_density(geom = "area", outline.type = "both")`. The geom's documentation lists which parameters it can accept. Inversely, when constructing a layer using a `⁠geom_()⁠` function, the `...` argument can be used to pass on parameters to the `stat` part of the layer. An example of this is `geom_area(stat = "density", adjust = 0.5)`. The stat's documentation lists which parameters it can accept. The `key_glyph` argument of `layer()` may also be passed on through `...`. This can be one of the functions described as key glyphs, to change the display of the layer in the legend.
`stat`	The statistical transformation to use on the data for this layer. When using a `⁠geom_*()⁠` function to construct a layer, the `stat` argument can be used the override the default coupling between geoms and stats. The `stat` argument accepts the following: A `Stat` ggproto subclass, for example `StatCount`. A string naming the stat. To give the stat as a string, strip the function name of the `stat_` prefix. For example, to use `stat_count()`, give the stat as `"count"`. For more information and other ways to specify the stat, see the layer stat documentation.

Examples


N1 <- Normal()
plot_pdf(N1) + geom_auc(to = -0.645)
plot_pdf(N1) + geom_auc(from = -0.645, to = 0.1, annotate = TRUE)

N2 <- Normal(0, c(1, 2))
plot_pdf(N2) + geom_auc(to = 0)
plot_pdf(N2) + geom_auc(from = -2, to = 2, annotate = TRUE)
N1 <- Normal()
plot_pdf(N1) + geom_auc(to = -0.645)
plot_pdf(N1) + geom_auc(from = -0.645, to = 0.1, annotate = TRUE)

N2 <- Normal(0, c(1, 2))
plot_pdf(N2) + geom_auc(to = 0)
plot_pdf(N2) + geom_auc(from = -2, to = 2, annotate = TRUE)

Create a Student's T distribution

Description

The Student's T distribution is closely related to the Normal() distribution, but has heavier tails. As $\nu$ increases to $\infty$ , the Student's T converges to a Normal. The T distribution appears repeatedly throughout classic frequentist hypothesis testing when comparing group means.

Usage

StudentsT(df)
StudentsT(df)

Arguments

`df`	Degrees of freedom. Can be any positive number. Often called $\nu$ in textbooks.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Students T random variable with df = $\nu$ .

Support: $R$ , the set of all real numbers

Mean: Undefined unless $\nu \ge 2$ , in which case the mean is zero.

Variance:

$\frac{\nu}{\nu - 2}$

Undefined if $\nu < 1$ , infinite when $1 < \nu \le 2$ .

Probability density function (p.d.f):

$f(x) = \frac{\Gamma(\frac{\nu + 1}{2})}{\sqrt{\nu \pi} \Gamma(\frac{\nu}{2})} (1 + \frac{x^2}{\nu} )^{- \frac{\nu + 1}{2}}$

Cumulative distribution function (c.d.f):

Nasty, omitted.

Moment generating function (m.g.f):

Undefined.

Value

A StudentsT object.

Examples


set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)
set.seed(27)

X <- StudentsT(3)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

### example: calculating p-values for two-sided T-test

# here the null hypothesis is H_0: mu = 3

# data to test
x <- c(3, 7, 11, 0, 7, 0, 4, 5, 6, 2)
nx <- length(x)

# calculate the T-statistic
t_stat <- (mean(x) - 3) / (sd(x) / sqrt(nx))
t_stat

# null distribution of statistic depends on sample size!
T <- StudentsT(df = nx - 1)

# calculate the two-sided p-value
1 - cdf(T, abs(t_stat)) + cdf(T, -abs(t_stat))

# exactly equivalent to the above
2 * cdf(T, -abs(t_stat))

# p-value for one-sided test
# H_0: mu <= 3   vs   H_A: mu > 3
1 - cdf(T, t_stat)

# p-value for one-sided test
# H_0: mu >= 3   vs   H_A: mu < 3
cdf(T, t_stat)

### example: calculating a 88 percent T CI for a mean

# lower-bound
mean(x) - quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# upper-bound
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# equivalent to
mean(x) + c(-1, 1) * quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

# also equivalent to
mean(x) + quantile(T, 0.12 / 2) * sd(x) / sqrt(nx)
mean(x) + quantile(T, 1 - 0.12 / 2) * sd(x) / sqrt(nx)

Compute the sufficient statistics of a distribution from data

Description

Generic function for computing the sufficient statistics of a distribution based on empirical data.

Usage

suff_stat(d, x, ...)
suff_stat(d, x, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`x`	A vector of data to compute the likelihood.
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

a named list of sufficient statistics

Examples

X <- Normal()
suff_stat(X, c(-1, 0, 0, 0, 3))
X <- Normal()
suff_stat(X, c(-1, 0, 0, 0, 3))

Compute the sufficient statistics for a Bernoulli distribution from data

Description

Compute the sufficient statistics for a Bernoulli distribution from data

Usage

## S3 method for class 'Bernoulli'
suff_stat(d, x, ...)
## S3 method for class 'Bernoulli'
suff_stat(d, x, ...)

Arguments

`d`	A `Bernoulli` object.
`x`	A vector of zeroes and ones.
`...`	Unused.

Value

A named list of the sufficient statistics of the Bernoulli distribution:

successes: The number of successful trials (sum(x == 1))
failures: The number of failed trials (sum(x == 0)).

Compute the sufficient statistics for the Binomial distribution from data

Description

Compute the sufficient statistics for the Binomial distribution from data

Usage

## S3 method for class 'Binomial'
suff_stat(d, x, ...)
## S3 method for class 'Binomial'
suff_stat(d, x, ...)

Arguments

`d`	A `Binomial` object.
`x`	A vector of zeroes and ones.
`...`	Unused.

Value

A named list of the sufficient statistics of the Binomial distribution:

successes: The total number of successful trials.
experiments: The number of experiments run.
trials: The number of trials run per experiment.

Compute the sufficient statistics of an Exponential distribution from data

Description

Compute the sufficient statistics of an Exponential distribution from data

Usage

## S3 method for class 'Exponential'
suff_stat(d, x, ...)
## S3 method for class 'Exponential'
suff_stat(d, x, ...)

Arguments

`d`	An `Exponential` object created by a call to `Exponential()`.
`x`	A vector of data.
`...`	Unused.

Value

A named list of the sufficient statistics of the exponential distribution:

sum: The sum of the observations.
samples: The number of observations.

Compute the sufficient statistics for a Gamma distribution from data

Description

sum: The sum of the data.
log_sum: The log of the sum of the data.
samples: The number of samples in the data.

Usage

## S3 method for class 'Gamma'
suff_stat(d, x, ...)
## S3 method for class 'Gamma'
suff_stat(d, x, ...)

Arguments

`d`	A `Gamma` object created by a call to `Gamma()`.
`x`	A vector to fit the Gamma distribution to.
`...`	Unused.

Value

a Gamma object

Compute the sufficient statistics for the Geometric distribution from data

Description

Compute the sufficient statistics for the Geometric distribution from data

Usage

## S3 method for class 'Geometric'
suff_stat(d, x, ...)
## S3 method for class 'Geometric'
suff_stat(d, x, ...)

Arguments

`d`	A `Geometric` object.
`x`	A vector of zeroes and ones.
`...`	Unused.

Value

A named list of the sufficient statistics of the Geometric distribution:

trials: The total number of trials ran until the first success.
experiments: The number of experiments run.

Compute the sufficient statistics for a Log-normal distribution from data

Description

Compute the sufficient statistics for a Log-normal distribution from data

Usage

## S3 method for class 'LogNormal'
suff_stat(d, x, ...)
## S3 method for class 'LogNormal'
suff_stat(d, x, ...)

Arguments

`d`	A `LogNormal` object created by a call to `LogNormal()`.
`x`	A vector of data.
`...`	Unused.

Value

A named list of the sufficient statistics of the normal distribution:

mu: The sample mean of the log of the data.
sigma: The sample standard deviation of the log of the data.
samples: The number of samples in the data.

Compute the sufficient statistics for a Normal distribution from data

Description

Compute the sufficient statistics for a Normal distribution from data

Usage

## S3 method for class 'Normal'
suff_stat(d, x, ...)
## S3 method for class 'Normal'
suff_stat(d, x, ...)

Arguments

`d`	A `Normal` object created by a call to `Normal()`.
`x`	A vector of data.
`...`	Unused.

Value

A named list of the sufficient statistics of the normal distribution:

mu: The sample mean of the data.
sigma: The sample standard deviation of the data.
samples: The number of samples in the data.

Compute the sufficient statistics of an Poisson distribution from data

Description

Compute the sufficient statistics of an Poisson distribution from data

Usage

## S3 method for class 'Poisson'
suff_stat(d, x, ...)
## S3 method for class 'Poisson'
suff_stat(d, x, ...)

Arguments

`d`	An `Poisson` object created by a call to `Poisson()`.
`x`	A vector of data.
`...`	Unused.

Value

A named list of the sufficient statistics of the Poisson distribution:

sum: The sum of the data.
samples: The number of samples in the data.

Return the support of a distribution

Description

Generic function for computing the support interval (minimum and maximum) for a given probability distribution object.

Usage

support(d, drop = TRUE, ...)
support(d, drop = TRUE, ...)

Arguments

`d`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Value

A vector (or matrix) with two elements (or columns) indicating the range (minimum and maximum) of the support.

Examples

X <- Normal()
support(X)
Y <- Uniform(-1, 1:3)
support(Y)
X <- Normal()
support(X)
Y <- Uniform(-1, 1:3)
support(Y)

Return the support of the Bernoulli distribution

Description

Return the support of the Bernoulli distribution

Usage

## S3 method for class 'Bernoulli'
support(d, drop = TRUE, ...)
## S3 method for class 'Bernoulli'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Bernoulli` object created by a call to `Bernoulli()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Beta distribution

Description

Return the support of the Beta distribution

Usage

## S3 method for class 'Beta'
support(d, drop = TRUE, ...)
## S3 method for class 'Beta'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Beta` object created by a call to `Beta()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Binomial distribution

Description

Return the support of the Binomial distribution

Usage

## S3 method for class 'Binomial'
support(d, drop = TRUE, ...)
## S3 method for class 'Binomial'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Binomial` object created by a call to `Binomial()`.
`drop`	logical. Shoul the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Cauchy distribution

Description

Return the support of the Cauchy distribution

Usage

## S3 method for class 'Cauchy'
support(d, drop = TRUE, ...)
## S3 method for class 'Cauchy'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Cauchy` object created by a call to `Cauchy()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the ChiSquare distribution

Description

Return the support of the ChiSquare distribution

Usage

## S3 method for class 'ChiSquare'
support(d, drop = TRUE, ...)
## S3 method for class 'ChiSquare'
support(d, drop = TRUE, ...)

Arguments

`d`	An `ChiSquare` object created by a call to `ChiSquare()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Erlang distribution

Description

Return the support of the Erlang distribution

Usage

## S3 method for class 'Erlang'
support(d, drop = TRUE, ...)
## S3 method for class 'Erlang'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Erlang` object created by a call to `Erlang()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Exponential distribution

Description

Return the support of the Exponential distribution

Usage

## S3 method for class 'Exponential'
support(d, drop = TRUE, ...)
## S3 method for class 'Exponential'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Exponential` object created by a call to `Exponential()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the FisherF distribution

Description

Return the support of the FisherF distribution

Usage

## S3 method for class 'FisherF'
support(d, drop = TRUE, ...)
## S3 method for class 'FisherF'
support(d, drop = TRUE, ...)

Arguments

`d`	An `FisherF` object created by a call to `FisherF()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Frechet distribution

Description

Return the support of the Frechet distribution

Usage

## S3 method for class 'Frechet'
support(d, drop = TRUE, ...)
## S3 method for class 'Frechet'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Frechet` object created by a call to `Frechet()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

In case of a single distribution object, a numeric vector of length 2 with the minimum and maximum value of the support (if drop = TRUE, default) or a matrix with 2 columns. In case of a vectorized distribution object, a matrix with 2 columns containing all minima and maxima.

Return the support of the Gamma distribution

Description

Return the support of the Gamma distribution

Usage

## S3 method for class 'Gamma'
support(d, drop = TRUE, ...)
## S3 method for class 'Gamma'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Gamma` object created by a call to `Gamma()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Geometric distribution

Description

Return the support of the Geometric distribution

Usage

## S3 method for class 'Geometric'
support(d, drop = TRUE, ...)
## S3 method for class 'Geometric'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Geometric` object created by a call to `Geometric()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of a GEV distribution

Description

Return the support of a GEV distribution

Usage

## S3 method for class 'GEV'
support(d, drop = TRUE, ...)
## S3 method for class 'GEV'
support(d, drop = TRUE, ...)

Arguments

`d`	An `GEV` object created by a call to `GEV()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

Return the support of the GP distribution

Description

Return the support of the GP distribution

Usage

## S3 method for class 'GP'
support(d, drop = TRUE, ...)
## S3 method for class 'GP'
support(d, drop = TRUE, ...)

Arguments

`d`	An `GP` object created by a call to `GP()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

Return the support of the Gumbel distribution

Description

Return the support of the Gumbel distribution

Usage

## S3 method for class 'Gumbel'
support(d, drop = TRUE, ...)
## S3 method for class 'Gumbel'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Gumbel` object created by a call to `Gumbel()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

Return the support of the hurdle negative binomial distribution

Description

Return the support of the hurdle negative binomial distribution

Usage

## S3 method for class 'HurdleNegativeBinomial'
support(d, drop = TRUE, ...)
## S3 method for class 'HurdleNegativeBinomial'
support(d, drop = TRUE, ...)

Arguments

`d`	An `HurdleNegativeBinomial` object created by a call to `HurdleNegativeBinomial()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the hurdle Poisson distribution

Description

Return the support of the hurdle Poisson distribution

Usage

## S3 method for class 'HurdlePoisson'
support(d, drop = TRUE, ...)
## S3 method for class 'HurdlePoisson'
support(d, drop = TRUE, ...)

Arguments

`d`	An `HurdlePoisson` object created by a call to `HurdlePoisson()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the HyperGeometric distribution

Description

Return the support of the HyperGeometric distribution

Usage

## S3 method for class 'HyperGeometric'
support(d, drop = TRUE, ...)
## S3 method for class 'HyperGeometric'
support(d, drop = TRUE, ...)

Arguments

`d`	An `HyperGeometric` object created by a call to `HyperGeometric()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Logistic distribution

Description

Return the support of the Logistic distribution

Usage

## S3 method for class 'Logistic'
support(d, drop = TRUE, ...)
## S3 method for class 'Logistic'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Logistic` object created by a call to `Logistic()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the LogNormal distribution

Description

Return the support of the LogNormal distribution

Usage

## S3 method for class 'LogNormal'
support(d, drop = TRUE, ...)
## S3 method for class 'LogNormal'
support(d, drop = TRUE, ...)

Arguments

`d`	An `LogNormal` object created by a call to `LogNormal()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the NegativeBinomial distribution

Description

Return the support of the NegativeBinomial distribution

Usage

## S3 method for class 'NegativeBinomial'
support(d, drop = TRUE, ...)
## S3 method for class 'NegativeBinomial'
support(d, drop = TRUE, ...)

Arguments

`d`	An `NegativeBinomial` object created by a call to `NegativeBinomial()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Normal distribution

Description

Return the support of the Normal distribution

Usage

## S3 method for class 'Normal'
support(d, drop = TRUE, ...)
## S3 method for class 'Normal'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Normal` object created by a call to `Normal()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

Return the support of the Poisson distribution

Description

Return the support of the Poisson distribution

Usage

## S3 method for class 'Poisson'
support(d, drop = TRUE, ...)
## S3 method for class 'Poisson'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Poisson` object created by a call to `Poisson()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the PoissonBinomial distribution

Description

Return the support of the PoissonBinomial distribution

Usage

## S3 method for class 'PoissonBinomial'
support(d, drop = TRUE, ...)
## S3 method for class 'PoissonBinomial'
support(d, drop = TRUE, ...)

Arguments

`d`	A `PoissonBinomial` object created by a call to `PoissonBinomial()`.
`drop`	logical. Shoul the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the RevWeibull distribution

Description

Return the support of the RevWeibull distribution

Usage

## S3 method for class 'RevWeibull'
support(d, drop = TRUE, ...)
## S3 method for class 'RevWeibull'
support(d, drop = TRUE, ...)

Arguments

`d`	An `RevWeibull` object created by a call to `RevWeibull()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the StudentsT distribution

Description

Return the support of the StudentsT distribution

Usage

## S3 method for class 'StudentsT'
support(d, drop = TRUE, ...)
## S3 method for class 'StudentsT'
support(d, drop = TRUE, ...)

Arguments

`d`	An `StudentsT` object created by a call to `StudentsT()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Tukey distribution

Description

Return the support of the Tukey distribution

Usage

## S3 method for class 'Tukey'
support(d, drop = TRUE, ...)
## S3 method for class 'Tukey'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Tukey` object created by a call to `Tukey()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Uniform distribution

Description

Return the support of the Uniform distribution

Usage

## S3 method for class 'Uniform'
support(d, drop = TRUE, ...)
## S3 method for class 'Uniform'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Uniform` object created by a call to `Uniform()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the Weibull distribution

Description

Return the support of the Weibull distribution

Usage

## S3 method for class 'Weibull'
support(d, drop = TRUE, ...)
## S3 method for class 'Weibull'
support(d, drop = TRUE, ...)

Arguments

`d`	An `Weibull` object created by a call to `Weibull()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the zero-inflated negative binomial distribution

Description

Return the support of the zero-inflated negative binomial distribution

Usage

## S3 method for class 'ZINegativeBinomial'
support(d, drop = TRUE, ...)
## S3 method for class 'ZINegativeBinomial'
support(d, drop = TRUE, ...)

Arguments

`d`	An `ZINegativeBinomial` object created by a call to `ZINegativeBinomial()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the zero-inflated Poisson distribution

Description

Return the support of the zero-inflated Poisson distribution

Usage

## S3 method for class 'ZIPoisson'
support(d, drop = TRUE, ...)
## S3 method for class 'ZIPoisson'
support(d, drop = TRUE, ...)

Arguments

`d`	An `ZIPoisson` object created by a call to `ZIPoisson()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the zero-truncated negative binomial distribution

Description

Return the support of the zero-truncated negative binomial distribution

Usage

## S3 method for class 'ZTNegativeBinomial'
support(d, drop = TRUE, ...)
## S3 method for class 'ZTNegativeBinomial'
support(d, drop = TRUE, ...)

Arguments

`d`	An `ZTNegativeBinomial` object created by a call to `ZTNegativeBinomial()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Return the support of the zero-truncated Poisson distribution

Description

Return the support of the zero-truncated Poisson distribution

Usage

## S3 method for class 'ZTPoisson'
support(d, drop = TRUE, ...)
## S3 method for class 'ZTPoisson'
support(d, drop = TRUE, ...)

Arguments

`d`	An `ZTPoisson` object created by a call to `ZTPoisson()`.
`drop`	logical. Should the result be simplified to a vector if possible?
`...`	Currently not used.

Value

A vector of length 2 with the minimum and maximum value of the support.

Create a Tukey distribution

Description

Tukey's studentized range distribution, used for Tukey's honestly significant differences test in ANOVA.

Usage

Tukey(nmeans, df, nranges)
Tukey(nmeans, df, nranges)

Arguments

`nmeans`	Sample size for each range.
`df`	Degrees of freedom.
`nranges`	Number of groups being compared.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

Support: $R^+$ , the set of positive real numbers.

Other properties of Tukey's Studentized Range Distribution are omitted, largely because the distribution is not fun to work with.

Value

A Tukey object.

Examples


set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Tukey(4L, 16L, 2L)
X

cdf(X, 4)
quantile(X, 0.7)

Create a Continuous Uniform distribution

Description

A distribution with constant density on an interval. The continuous analogue to the Categorical() distribution.

Usage

Uniform(a = 0, b = 1)
Uniform(a = 0, b = 1)

Arguments

`a`	The a parameter. `a` can be any value in the set of real numbers. Defaults to `0`.
`b`	The a parameter. `b` can be any value in the set of real numbers. It should be strictly bigger than `a`, but if is not, the order of the parameters is inverted. Defaults to `1`.

Value

A Uniform object.

Examples


set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))
set.seed(27)

X <- Uniform(1, 2)
X

random(X, 10)

pdf(X, 0.7)
log_pdf(X, 0.7)

cdf(X, 0.7)
quantile(X, 0.7)

cdf(X, quantile(X, 0.7))
quantile(X, cdf(X, 0.7))

Compute the moments of a probability distribution

Description

Generic functions for computing moments (variance, skewness, excess kurtosis) from probability distributions.

Usage

variance(x, ...)

skewness(x, ...)

kurtosis(x, ...)
variance(x, ...)

skewness(x, ...)

kurtosis(x, ...)

Arguments

`x`	An object. The package provides methods for distribution objects such as those from `Normal()` or `Binomial()` etc.
`...`	Arguments passed to methods. Unevaluated arguments will generate a warning to catch mispellings or other possible errors.

Details

The functions variance, skewness, and kurtosis are new generic functions for computing moments of probability distributions such as those provided in this package. Additionally, the probability distributions from distributions3 all have methods for the mean generic. Moreover, quantiles can be computed with methods for quantile. For examples illustrating the usage with probability distribution objects, see the manual pages of the respective distributions, e.g., Normal or Binomial etc.

Value

Numeric vector with the values of the moments.

Create a Weibull distribution

Description

Generalization of the gamma distribution. Often used in survival and time-to-event analyses.

Usage

Weibull(shape, scale)
Weibull(shape, scale)

Arguments

`shape`	The shape parameter $k$ . Can be any positive real number.
`scale`	The scale parameter $\lambda$ . Can be any positive real number.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail and much greater clarity.

In the following, let $X$ be a Weibull random variable with success probability p = $p$ .

Support: $R^+$ and zero.

Mean: $\lambda \Gamma(1+1/k)$ , where $\Gamma$ is the gamma function.

Variance: $\lambda [ \Gamma (1 + \frac{2}{k} ) - (\Gamma(1+ \frac{1}{k}))^2 ]$

Probability density function (p.d.f):

$f(x) = \frac{k}{\lambda}(\frac{x}{\lambda})^{k-1}e^{-(x/\lambda)^k}, x \ge 0$

Cumulative distribution function (c.d.f):

$F(x) = 1 - e^{-(x/\lambda)^k}, x \ge 0$

Moment generating function (m.g.f):

$\sum_{n=0}^\infty \frac{t^n\lambda^n}{n!} \Gamma(1+n/k), k \ge 1$

Value

A Weibull object.

Examples


set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)
set.seed(27)

X <- Weibull(0.3, 2)
X

random(X, 10)

pdf(X, 2)
log_pdf(X, 2)

cdf(X, 4)
quantile(X, 0.7)

Create a zero-inflated negative binomial distribution

Description

Zero-inflated negative binomial distributions are frequently used to model counts with overdispersion and many zero observations.

Usage

ZINegativeBinomial(mu, theta, pi)
ZINegativeBinomial(mu, theta, pi)

Arguments

`mu`	Location parameter of the negative binomial component of the distribution. Can be any positive number.
`theta`	Overdispersion parameter of the negative binomial component of the distribution. Can be any positive number.
`pi`	Zero-inflation probability, can be any value in `⁠[0, 1]⁠`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a zero-inflated negative binomial random variable with parameters mu = $\mu$ and theta = $\theta$ .

Support: $\{0, 1, 2, 3, ...\}$

Mean: $(1 - \pi) \cdot \mu$

Variance: $(1 - \pi) \cdot \mu \cdot (1 + (\pi + 1/\theta) \cdot \mu)$

Probability mass function (p.m.f.):

$P(X = k) = \pi \cdot I_{0}(k) + (1 - \pi) \cdot f(k; \mu, \theta)$

where $I_{0}(k)$ is the indicator function for zero and $f(k; \mu, \theta)$ is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.):

$P(X \le k) = \pi + (1 - \pi) \cdot F(k; \mu, \theta)$

where $F(k; \mu, \theta)$ is the c.d.f. of the NegativeBinomial distribution.

Moment generating function (m.g.f.):

Omitted for now.

Value

A ZINegativeBinomial object.

Examples

## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated negative binomial distribution
X <- ZINegativeBinomial(mu = 2.5, theta = 1, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a zero-inflated Poisson distribution

Description

Zero-inflated Poisson distributions are frequently used to model counts with many zero observations.

Usage

ZIPoisson(lambda, pi)
ZIPoisson(lambda, pi)

Arguments

`lambda`	Parameter of the Poisson component of the distribution. Can be any positive number.
`pi`	Zero-inflation probability, can be any value in `⁠[0, 1]⁠`.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a zero-inflated Poisson random variable with parameter lambda = $\lambda$ .

Support: $\{0, 1, 2, 3, ...\}$

Mean: $(1 - \pi) \cdot \lambda$

Variance: $(1 - \pi) \cdot \lambda \cdot (1 + \pi \cdot \lambda)$

Probability mass function (p.m.f.):

$P(X = k) = \pi \cdot I_{0}(k) + (1 - \pi) \cdot f(k; \lambda)$

where $I_{0}(k)$ is the indicator function for zero and $f(k; \lambda)$ is the p.m.f. of the Poisson distribution.

Cumulative distribution function (c.d.f.):

$P(X \le k) = \pi + (1 - \pi) \cdot F(k; \lambda)$

where $F(k; \lambda)$ is the c.d.f. of the Poisson distribution.

Moment generating function (m.g.f.):

$E(e^{tX}) = \pi + (1 - \pi) \cdot e^{\lambda (e^t - 1)}$

Value

A ZIPoisson object.

Examples

## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-inflated Poisson distribution
X <- ZIPoisson(lambda = 2.5, pi = 0.25)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a zero-truncated negative binomial distribution

Description

Zero-truncated negative binomial distributions are frequently used to model counts where zero observations cannot occur or have been excluded.

Usage

ZTNegativeBinomial(mu, theta)
ZTNegativeBinomial(mu, theta)

Arguments

`mu`	Location parameter of the negative binomial component of the distribution. Can be any positive number.
`theta`	Overdispersion parameter of the negative binomial component of the distribution. Can be any positive number.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a zero-truncated negative binomial random variable with parameter mu = $\mu$ .

Support: $\{1, 2, 3, ...\}$

Mean:

$\mu \cdot \frac{1}{1 - F(0; \mu, \theta)}$

where $F(k; \mu, \theta)$ is the c.d.f. of the NegativeBinomial distribution.

Variance: $m \cdot (\mu + 1 - m)$ , where $m$ is the mean above.

Probability mass function (p.m.f.):

$P(X = k) = \frac{f(k; \mu, \theta)}{1 - F(0; \mu, \theta)}$

where $f(k; \mu, \theta)$ is the p.m.f. of the NegativeBinomial distribution.

Cumulative distribution function (c.d.f.):

$P(X = k) = \frac{F(k; \mu, \theta)}{1 - F(0; \mu, \theta)}$

Moment generating function (m.g.f.):

Omitted for now.

Value

A ZTNegativeBinomial object.

Examples

## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated negative binomial distribution
X <- ZTNegativeBinomial(mu = 2.5, theta = 1)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Create a zero-truncated Poisson distribution

Description

Zero-truncated Poisson distributions are frequently used to model counts where zero observations cannot occur or have been excluded.

Usage

ZTPoisson(lambda)
ZTPoisson(lambda)

Arguments

lambda

Parameter of the underlying untruncated Poisson distribution. Can be any positive number.

Details

We recommend reading this documentation on https://alexpghayes.github.io/distributions3/, where the math will render with additional detail.

In the following, let $X$ be a zero-truncated Poisson random variable with parameter lambda = $\lambda$ .

Support: $\{1, 2, 3, ...\}$

Mean:

$\lambda \cdot \frac{1}{1 - e^{-\lambda}}$

Variance: $m \cdot (\lambda + 1 - m)$ , where $m$ is the mean above.

Probability mass function (p.m.f.):

$P(X = k) = \frac{f(k; \lambda)}{1 - f(0; \lambda)}$

where $f(k; \lambda)$ is the p.m.f. of the Poisson distribution.

Cumulative distribution function (c.d.f.):

$P(X = k) = \frac{F(k; \lambda)}{1 - F(0; \lambda)}$

where $F(k; \lambda)$ is the c.d.f. of the Poisson distribution.

Moment generating function (m.g.f.):

$E(e^{tX}) = \frac{1}{1 - e^{-\lambda}} \cdot e^{\lambda (e^t - 1)}$

Value

A ZTPoisson object.

Examples

## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)
## set up a zero-truncated Poisson distribution
X <- ZTPoisson(lambda = 2.5)
X

## standard functions
pdf(X, 0:8)
cdf(X, 0:8)
quantile(X, seq(0, 1, by = 0.25))

## cdf() and quantile() are inverses for each other
quantile(X, cdf(X, 3))

## density visualization
plot(0:8, pdf(X, 0:8), type = "h", lwd = 2)

## corresponding sample with histogram of empirical frequencies
set.seed(0)
x <- random(X, 500)
hist(x, breaks = -1:max(x) + 0.5)

Package 'distributions3'

Help Index

Utilities for distributions3 objects

Description

Usage

Arguments

Examples

Create a Bernoulli distribution

Description

Usage

Arguments

Details

Value

See Also

Examples

Create a Beta distribution

Description

Usage

Arguments

Value

See Also

Examples

Create a Binomial distribution

Description

Usage

Arguments

Details

Value

See Also

Examples

Create a Categorical distribution

Description

Usage

Arguments

Value

See Also

Examples

Create a Cauchy distribution

Description

Usage

Arguments

Details

Value

See Also

Examples

Evaluate the cumulative distribution function of a probability distribution

Description

Usage

Arguments

Value

Examples

Evaluate the cumulative distribution function of a Bernoulli distribution

Description

Usage

Arguments

Value

Examples

Evaluate the cumulative distribution function of a Beta distribution

Description

Usage

Arguments

Value

Examples

Evaluate the cumulative distribution function of a Binomial distribution

Description

Usage

Arguments

Value

Examples

Evaluate the cumulative distribution function of a Categorical distribution

Description

Usage

Arguments

Value

Examples

Evaluate the cumulative distribution function of a Cauchy distribution

Description

Usage

Arguments

Value

Utilities for `distributions3` objects