Title:	Vectorised Probability Distributions
Version:	0.6.0
Description:	Vectorised distribution objects with tools for manipulating, visualising, and using probability distributions. Designed to allow model prediction outputs to return distributions rather than their parameters, allowing users to directly interact with predictive distributions in a data-oriented workflow. In addition to providing generic replacements for p/d/q/r functions, other useful statistics can be computed including means, variances, intervals, and highest density regions.
License:	GPL-3
Depends:	R (≥ 4.0.0)
Imports:	vctrs (≥ 0.3.0), rlang (≥ 0.4.5), generics, stats, numDeriv, utils, lifecycle, pillar
Suggests:	testthat (≥ 2.1.0), covr, mvtnorm, actuar (≥ 2.0.0), evd, ggdist, ggplot2, gk, pkgdown
RdMacros:	lifecycle
URL:	https://pkg.mitchelloharawild.com/distributional/, https://github.com/mitchelloharawild/distributional
BugReports:	https://github.com/mitchelloharawild/distributional/issues
Encoding:	UTF-8
Language:	en-GB
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-01-14 10:04:53 UTC; mitchell
Author:	Mitchell O'Hara-Wild [aut, cre], Matthew Kay [aut], Alex Hayes [aut], Rob Hyndman [aut], Earo Wang [ctb], Vencislav Popov [ctb]
Maintainer:	Mitchell O'Hara-Wild <mail@mitchelloharawild.com>
Repository:	CRAN
Date/Publication:	2026-01-14 10:50:03 UTC

distributional: Vectorised Probability Distributions

Description

Vectorised distribution objects with tools for manipulating, visualising, and using probability distributions. Designed to allow model prediction outputs to return distributions rather than their parameters, allowing users to directly interact with predictive distributions in a data-oriented workflow. In addition to providing generic replacements for p/d/q/r functions, other useful statistics can be computed including means, variances, intervals, and highest density regions.

Author(s)

Maintainer: Mitchell O'Hara-Wild mail@mitchelloharawild.com (ORCID)

Authors:

• Matthew Kay (ORCID)

• Alex Hayes (ORCID)

• Rob Hyndman (ORCID)

Other contributors:

• Earo Wang (ORCID) [contributor]

• Vencislav Popov (ORCID) [contributor]

See Also

Useful links:

• https://pkg.mitchelloharawild.com/distributional/

• https://github.com/mitchelloharawild/distributional

• Report bugs at https://github.com/mitchelloharawild/distributional/issues

The cumulative distribution function

Description

Usage

cdf(x, q, ..., log = FALSE)

## S3 method for class 'distribution'
cdf(x, q, ...)

Arguments

Covariance

Description

A generic function for computing the covariance of an object.

Usage

covariance(x, ...)

Arguments

See Also

covariance.distribution(), variance()

Covariance of a probability distribution

Description

Returns the empirical covariance of the probability distribution. If the method does not exist, the covariance of a random sample will be returned.

Usage

## S3 method for class 'distribution'
covariance(x, ...)

Arguments

The probability density/mass function

Description

Computes the probability density function for a continuous distribution, or the probability mass function for a discrete distribution.

Usage

## S3 method for class 'distribution'
density(x, at, ..., log = FALSE)

Arguments

The 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

dist_bernoulli(prob)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_bernoulli.html

In the following, let X be a Bernoulli random variable with parameter prob = 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

Skewness:

\frac{1 - 2p}{\sqrt{p(1-p)}} = \frac{q - p}{\sqrt{pq}}

Excess Kurtosis:

\frac{1 - 6p(1-p)}{p(1-p)} = \frac{1 - 6pq}{pq}

See Also

stats::Binomial

Examples

dist <- dist_bernoulli(prob = c(0.05, 0.5, 0.3, 0.9, 0.1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Beta distribution

Description

The Beta distribution is a continuous probability distribution defined on the interval [0, 1], commonly used to model probabilities and proportions.

Usage

dist_beta(shape1, shape2)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_beta.html

In the following, let X be a Beta random variable with parameters shape1 = \alpha and shape2 = \beta.

Support: x \in [0, 1]

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

Variance: \frac{\alpha\beta}{(\alpha + \beta)^2(\alpha + \beta + 1)}

Probability density function (p.d.f):

f(x) = \frac{x^{\alpha - 1}(1-x)^{\beta - 1}}{B(\alpha, \beta)} = \frac{\Gamma(\alpha + \beta)}{\Gamma(\alpha)\Gamma(\beta)} x^{\alpha - 1}(1-x)^{\beta - 1}

where B(\alpha, \beta) = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha + \beta)} is the Beta function.

Cumulative distribution function (c.d.f):

F(x) = I_x(alpha, beta) = \frac{B(x; \alpha, \beta)}{B(\alpha, \beta)}

where I_x(\alpha, \beta) is the regularized incomplete beta function and B(x; \alpha, \beta) = \int_0^x t^{\alpha-1}(1-t)^{\beta-1} dt.

Moment generating function (m.g.f):

The moment generating function does not have a simple closed form, but the moments can be calculated as:

E(X^k) = \prod_{r=0}^{k-1} \frac{\alpha + r}{\alpha + \beta + r}

See Also

stats::Beta

Examples

dist <- dist_beta(shape1 = c(0.5, 5, 1, 2, 2), shape2 = c(0.5, 1, 3, 2, 5))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The 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

dist_binomial(size, prob)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_binomial.html

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.

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

Skewness:

\frac{1 - 2p}{\sqrt{np(1-p)}}

Excess kurtosis:

\frac{1 - 6p(1-p)}{np(1-p)}

See Also

stats::Binomial

Examples

dist <- dist_binomial(size = 1:5, prob = c(0.05, 0.5, 0.3, 0.9, 0.1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Burr distribution

Description

The Burr distribution (Type XII) is a flexible continuous probability distribution often used for modeling income distributions, reliability data, and failure times.

Usage

dist_burr(shape1, shape2, rate = 1, scale = 1/rate)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_burr.html

In the following, let X be a Burr random variable with parameters shape1 = \alpha, shape2 = \gamma, and rate = \lambda.

Support: x \in (0, \infty)

Mean: \frac{\lambda^{-1/\alpha} \gamma B(\gamma - 1/\alpha, 1 + 1/\alpha)}{\gamma} (for \alpha \gamma > 1)

Variance: \frac{\lambda^{-2/\alpha} \gamma B(\gamma - 2/\alpha, 1 + 2/\alpha)}{\gamma} - \mu^2 (for \alpha \gamma > 2)

Probability density function (p.d.f):

f(x) = \alpha \gamma \lambda x^{\alpha - 1} (1 + \lambda x^\alpha)^{-\gamma - 1}

Cumulative distribution function (c.d.f):

F(x) = 1 - (1 + \lambda x^\alpha)^{-\gamma}

Quantile function:

F^{-1}(p) = \lambda^{-1/\alpha} ((1 - p)^{-1/\gamma} - 1)^{1/\alpha}

Moment generating function (m.g.f):

Does not exist in closed form.

See Also

actuar::Burr

Examples

dist <- dist_burr(shape1 = c(1,1,1,2,3,0.5), shape2 = c(1,2,3,1,1,2))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Categorical distribution

Description

Categorical distributions are used to represent events with multiple outcomes, such as what number appears on the roll of a dice. This is also referred to as the 'generalised Bernoulli' or 'multinoulli' distribution. The Categorical distribution is a special case of the Multinomial() distribution with n = 1.

Usage

dist_categorical(prob, outcomes = NULL)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_categorical.html

In the following, let X be a Categorical random variable with probability parameters prob = \{p_1, p_2, \ldots, p_k\}.

The Categorical probability distribution is widely used to model the occurance of multiple events. A simple example is the roll of a dice, where p = \{1/6, 1/6, 1/6, 1/6, 1/6, 1/6\} giving equal chance of observing each number on a 6 sided dice.

Support: \{1, \ldots, k\}

Mean: Not defined for unordered categories. For ordered categories with integer outcomes \{1, 2, \ldots, k\}, the mean is:

E(X) = \sum_{i=1}^{k} i \cdot p_i

Variance: Not defined for unordered categories. For ordered categories with integer outcomes \{1, 2, \ldots, k\}, the variance is:

\text{Var}(X) = \sum_{i=1}^{k} i^2 \cdot p_i - \left(\sum_{i=1}^{k} i \cdot p_i\right)^2

Probability mass function (p.m.f):

P(X = i) = p_i

Cumulative distribution function (c.d.f):

The c.d.f is undefined for unordered categories. For ordered categories with outcomes x_1 < x_2 < \ldots < x_k, the c.d.f is:

P(X \le x_j) = \sum_{i=1}^{j} p_i

Moment generating function (m.g.f):

E(e^{tX}) = \sum_{i=1}^{k} e^{tx_i} \cdot p_i

Skewness: Approximated numerically for ordered categories.

Kurtosis: Approximated numerically for ordered categories.

See Also

stats::Multinomial

Examples

dist <- dist_categorical(prob = list(c(0.05, 0.5, 0.15, 0.2, 0.1), c(0.3, 0.1, 0.6)))

dist

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

# The outcomes aren't ordered, so many statistics are not applicable.
cdf(dist, 0.6)
quantile(dist, 0.7)
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

# Some of these statistics are meaningful for ordered outcomes
dist <- dist_categorical(list(rpois(26, 3)), list(ordered(letters)))
dist
cdf(dist, "m")
quantile(dist, 0.5)

dist <- dist_categorical(
  prob = list(c(0.05, 0.5, 0.15, 0.2, 0.1), c(0.3, 0.1, 0.6)),
  outcomes = list(letters[1:5], letters[24:26])
)

generate(dist, 10)

density(dist, "a")
density(dist, "z", log = TRUE)

The Cauchy distribution

Description

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

dist_cauchy(location, scale)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_cauchy.html

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.

See Also

stats::Cauchy

Examples

dist <- dist_cauchy(location = c(0, 0, 0, -2), scale = c(0.5, 1, 2, 1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The (non-central) Chi-Squared 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

dist_chisq(df, ncp = 0)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_chisq.html

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

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

Mean: k + \lambda

Variance: 2(k + 2\lambda)

Probability density function (p.d.f):

For the central chi-squared distribution (\lambda = 0):

f(x) = \frac{1}{2^{k/2} \Gamma(k/2)} x^{k/2 - 1} e^{-x/2}

For the non-central chi-squared distribution (\lambda > 0):

f(x) = \frac{1}{2} e^{-(x+\lambda)/2} \left(\frac{x}{\lambda}\right)^{k/4-1/2} I_{k/2-1}\left(\sqrt{\lambda x}\right)

where I_\nu(z) is the modified Bessel function of the first kind.

Cumulative distribution function (c.d.f):

For the central chi-squared distribution (\lambda = 0):

F(x) = \frac{\gamma(k/2, x/2)}{\Gamma(k/2)} = P(k/2, x/2)

where \gamma(s, x) is the lower incomplete gamma function and P(s, x) is the regularized gamma function.

For the non-central chi-squared distribution (\lambda > 0):

F(x) = \sum_{j=0}^{\infty} \frac{e^{-\lambda/2} (\lambda/2)^j}{j!} P(k/2 + j, x/2)

This is approximated numerically.

Moment generating function (m.g.f):

For the central chi-squared distribution (\lambda = 0):

E(e^{tX}) = (1 - 2t)^{-k/2}, \quad t < 1/2

For the non-central chi-squared distribution (\lambda > 0):

E(e^{tX}) = \frac{e^{\lambda t / (1 - 2t)}}{(1 - 2t)^{k/2}}, \quad t < 1/2

Skewness:

\gamma_1 = \frac{2^{3/2}(k + 3\lambda)}{(k + 2\lambda)^{3/2}}

For the central case (\lambda = 0), this simplifies to \sqrt{8/k}.

Excess Kurtosis:

\gamma_2 = \frac{12(k + 4\lambda)}{(k + 2\lambda)^2}

For the central case (\lambda = 0), this simplifies to 12/k.

See Also

stats::Chisquare

Examples

dist <- dist_chisq(df = c(1,2,3,4,6,9))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The degenerate distribution

Description

The degenerate distribution takes a single value which is certain to be observed. It takes a single parameter, which is the value that is observed by the distribution.

Usage

dist_degenerate(x)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_degenerate.html

In the following, let X be a degenerate random variable with value x = k_0.

Support: \{k_0\}, a single point

Mean: \mu = k_0

Variance: \sigma^2 = 0

Probability density function (p.d.f):

f(x) = 1 \textrm{ for } x = k_0

f(x) = 0 \textrm{ for } x \neq k_0

Cumulative distribution function (c.d.f):

F(t) = 0 \textrm{ for } t < k_0

F(t) = 1 \textrm{ for } t \ge k_0

Moment generating function (m.g.f):

E(e^{tX}) = e^{k_0 t}

Skewness: Undefined (NA)

Excess Kurtosis: Undefined (NA)

See Also

stats::Distributions

Examples

dist <- dist_degenerate(x = 1:5)

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Exponential Distribution

Description

Exponential distributions are frequently used to model waiting times and the time between events in a Poisson process.

Usage

dist_exponential(rate)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_exponential.html

In the following, let X be an Exponential random variable with 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):

E(e^{tX}) = \frac{\lambda}{\lambda - t}, \quad t < \lambda

See Also

stats::Exponential

Examples

dist <- dist_exponential(rate = c(2, 1, 2/3))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The F Distribution

Description

The F distribution is commonly used in statistical inference, particularly in the analysis of variance (ANOVA), testing the equality of variances, and in regression analysis. It arises as the ratio of two scaled chi-squared distributions divided by their respective degrees of freedom.

Usage

dist_f(df1, df2, ncp = NULL)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_f.html

In the following, let X be an F random variable with numerator degrees of freedom df1 = d_1 and denominator degrees of freedom df2 = d_2.

Support: x \in (0, \infty)

Mean:

For the central F distribution (ncp = NULL):

E(X) = \frac{d_2}{d_2 - 2}

for d_2 > 2, otherwise undefined.

For the non-central F distribution with non-centrality parameter ncp = \lambda:

E(X) = \frac{d_2 (d_1 + \lambda)}{d_1 (d_2 - 2)}

for d_2 > 2, otherwise undefined.

Variance:

For the central F distribution (ncp = NULL):

\text{Var}(X) = \frac{2 d_2^2 (d_1 + d_2 - 2)}{d_1 (d_2 - 2)^2 (d_2 - 4)}

for d_2 > 4, otherwise undefined.

For the non-central F distribution with non-centrality parameter ncp = \lambda:

\text{Var}(X) = \frac{2 d_2^2}{d_1^2} \cdot \frac{(d_1 + \lambda)^2 + (d_1 + 2\lambda)(d_2 - 2)}{(d_2 - 2)^2 (d_2 - 4)}

for d_2 > 4, otherwise undefined.

Skewness:

For the central F distribution (ncp = NULL):

\text{Skew}(X) = \frac{(2 d_1 + d_2 - 2) \sqrt{8 (d_2 - 4)}}{(d_2 - 6) \sqrt{d_1 (d_1 + d_2 - 2)}}

for d_2 > 6, otherwise undefined.

For the non-central F distribution, skewness has no simple closed form and is not computed.

Excess Kurtosis:

For the central F distribution (ncp = NULL):

\text{Kurt}(X) = \frac{12[d_1 (5 d_2 - 22)(d_1 + d_2 - 2) + (d_2 - 4)(d_2 - 2)^2]}{d_1 (d_2 - 6)(d_2 - 8)(d_1 + d_2 - 2)}

for d_2 > 8, otherwise undefined.

For the non-central F distribution, kurtosis has no simple closed form and is not computed.

Probability density function (p.d.f):

For the central F distribution (ncp = NULL):

f(x) = \frac{\sqrt{\frac{(d_1 x)^{d_1} d_2^{d_2}}{(d_1 x + d_2)^{d_1 + d_2}}}}{x \, B(d_1/2, d_2/2)}

where B(\cdot, \cdot) is the beta function.

For the non-central F distribution, the density involves an infinite series and is approximated numerically.

Cumulative distribution function (c.d.f):

The c.d.f. does not have a simple closed form expression and is approximated numerically using regularized incomplete beta functions and related special functions.

Moment generating function (m.g.f):

The moment generating function for the F distribution does not exist in general (it diverges for t > 0).

See Also

stats::FDist

Examples

dist <- dist_f(df1 = c(1,2,5,10,100), df2 = c(1,1,2,1,100))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The 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

dist_gamma(shape, rate = 1/scale, scale = 1/rate)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gamma.html

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

See Also

stats::GammaDist

Examples

dist <- dist_gamma(shape = c(1,2,3,5,9,7.5,0.5), rate = c(0.5,0.5,0.5,1,2,1,1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Geometric Distribution

Description

The Geometric distribution can be thought of as a generalization of the dist_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

dist_geometric(prob)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_geometric.html

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

Support: \{0, 1, 2, 3, ...\}

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

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

Probability mass function (p.m.f):

P(X = k) = p(1-p)^k

Cumulative distribution function (c.d.f):

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

Moment generating function (m.g.f):

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

Skewness:

\frac{2 - p}{\sqrt{1 - p}}

Excess Kurtosis:

6 + \frac{p^2}{1 - p}

See Also

stats::Geometric

Examples

dist <- dist_geometric(prob = c(0.2, 0.5, 0.8))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Generalized Extreme Value Distribution

Description

The GEV distribution is widely used in extreme value theory to model the distribution of maxima (or minima) of samples. The parametric form encompasses the Gumbel, Frechet, and reverse Weibull distributions.

Usage

dist_gev(location, scale, shape)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gev.html

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

Support:

• x \in \mathbb{R} (all real numbers) if \xi = 0

• x \geq \mu - \sigma/\xi if \xi > 0

• x \leq \mu - \sigma/\xi if \xi < 0

Mean:

E(X) = \begin{cases} \mu + \sigma \gamma & \text{if } \xi = 0 \\ \mu + \sigma \frac{\Gamma(1-\xi) - 1}{\xi} & \text{if } \xi < 1 \\ \infty & \text{if } \xi \geq 1 \end{cases}

where \gamma \approx 0.5772 is the Euler-Mascheroni constant and \Gamma(\cdot) is the gamma function.

Median:

\text{Median}(X) = \begin{cases} \mu - \sigma \log(\log 2) & \text{if } \xi = 0 \\ \mu + \sigma \frac{(\log 2)^{-\xi} - 1}{\xi} & \text{if } \xi \neq 0 \end{cases}

Variance:

\text{Var}(X) = \begin{cases} \frac{\pi^2 \sigma^2}{6} & \text{if } \xi = 0 \\ \frac{\sigma^2}{\xi^2} [\Gamma(1-2\xi) - \Gamma(1-\xi)^2] & \text{if } \xi < 0.5 \\ \infty & \text{if } \xi \geq 0.5 \end{cases}

Probability density function (p.d.f):

For \xi = 0 (Gumbel):

f(x) = \frac{1}{\sigma} \exp\left(-\frac{x-\mu}{\sigma}\right) \exp\left[-\exp\left(-\frac{x-\mu}{\sigma}\right)\right]

For \xi \neq 0:

f(x) = \frac{1}{\sigma} \left[1 + \xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi-1} \exp\left\{-\left[1 + \xi\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}

where 1 + \xi(x-\mu)/\sigma > 0.

Cumulative distribution function (c.d.f):

For \xi = 0 (Gumbel):

F(x) = \exp\left[-\exp\left(-\frac{x-\mu}{\sigma}\right)\right]

For \xi \neq 0:

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

where 1 + \xi(x-\mu)/\sigma > 0.

Quantile function:

For \xi = 0 (Gumbel):

Q(p) = \mu - \sigma \log(-\log p)

For \xi \neq 0:

Q(p) = \mu + \frac{\sigma}{\xi}\left[(-\log p)^{-\xi} - 1\right]

References

Jenkinson, A. F. (1955) The frequency distribution of the annual maximum (or minimum) of meteorological elements. Quart. J. R. Met. Soc., 81, 158–171.

See Also

evd::dgev()

Examples

# Create GEV distributions with different shape parameters

# Gumbel distribution (shape = 0)
gumbel <- dist_gev(location = 0, scale = 1, shape = 0)

# Frechet distribution (shape > 0, heavy-tailed)
frechet <- dist_gev(location = 0, scale = 1, shape = 0.3)

# Reverse Weibull distribution (shape < 0, bounded above)
weibull <- dist_gev(location = 0, scale = 1, shape = -0.2)

dist <- c(gumbel, frechet, weibull)
dist

# Statistical properties
mean(dist)
median(dist)
variance(dist)

# Generate random samples
generate(dist, 10)

# Evaluate density
density(dist, 2)
density(dist, 2, log = TRUE)

# Evaluate cumulative distribution
cdf(dist, 4)

# Calculate quantiles
quantile(dist, 0.95)

The generalised g-and-h Distribution

Description

The generalised g-and-h distribution is a flexible distribution used to model univariate data, similar to the g-k distribution. It is known for its ability to handle skewness and heavy-tailed behavior.

Usage

dist_gh(A, B, g, h, c = 0.8)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gh.html

In the following, let X be a g-and-h random variable with parameters A = A, B = B, g = g, h = h, and c = c.

Support: (-\infty, \infty)

Mean: Does not have a closed-form expression. Approximated numerically.

Variance: Does not have a closed-form expression. Approximated numerically.

Probability density function (p.d.f):

The g-and-h distribution does not have a closed-form expression for its density. The density is approximated numerically from the quantile function. The distribution is defined through its quantile function:

Q(u) = A + B \left( 1 + c \frac{1 - \exp(-gz(u))}{1 + \exp(-gz(u))} \right) \exp(h z(u)^2/2) z(u)

where z(u) = \Phi^{-1}(u) is the standard normal quantile function.

Cumulative distribution function (c.d.f):

Does not have a closed-form expression. The cumulative distribution function is approximated numerically by inverting the quantile function.

Quantile function:

Q(p) = A + B \left( 1 + c \frac{1 - \exp(-g\Phi^{-1}(p))}{1 + \exp(-g\Phi^{-1}(p))} \right) \exp(h (\Phi^{-1}(p))^2/2) \Phi^{-1}(p)

where \Phi^{-1}(p) is the standard normal quantile function.

See Also

gk::dgh(), gk::pgh(), gk::qgh(), gk::rgh(), dist_gk()

Examples

dist <- dist_gh(A = 0, B = 1, g = 0, h = 0.5)
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The g-and-k Distribution

Description

The g-and-k distribution is a flexible distribution often used to model univariate data. It is particularly known for its ability to handle skewness and heavy-tailed behavior.

Usage

dist_gk(A, B, g, k, c = 0.8)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gk.html

In the following, let X be a g-k random variable with parameters A, B, g, k, and c.

Support: (-\infty, \infty)

Mean: Not available in closed form.

Variance: Not available in closed form.

Probability density function (p.d.f):

The g-k distribution does not have a closed-form expression for its density. Instead, it is defined through its quantile function:

Q(u) = A + B \left( 1 + c \frac{1 - \exp(-gz(u))}{1 + \exp(-gz(u))} \right) (1 + z(u)^2)^k z(u)

where z(u) = \Phi^{-1}(u), the standard normal quantile of u.

Cumulative distribution function (c.d.f):

The cumulative distribution function is typically evaluated numerically due to the lack of a closed-form expression.

See Also

gk::dgk, dist_gh

Examples

dist <- dist_gk(A = 0, B = 1, g = 0, k = 0.5)
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Generalized Pareto Distribution

Description

The GPD distribution is commonly used to model the tails of distributions, particularly in extreme value theory.

The Pickands–Balkema–De Haan theorem states that for a large class of distributions, the tail (above some threshold) can be approximated by a GPD.

Usage

dist_gpd(location, scale, shape)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gpd.html

In the following, let X be a Generalized Pareto random variable with parameters location = a, scale = b > 0, and shape = s.

Support: x \ge a if s \ge 0, a \le x \le a - b/s if s < 0

Mean:

E(X) = a + \frac{b}{1 - s} \quad \textrm{for } s < 1

E(X) = \infty for s \ge 1

Variance:

\textrm{Var}(X) = \frac{b^2}{(1-s)^2(1-2s)} \quad \textrm{for } s < 0.5

\textrm{Var}(X) = \infty for s \ge 0.5

Probability density function (p.d.f):

For s = 0:

f(x) = \frac{1}{b}\exp\left(-\frac{x-a}{b}\right) \quad \textrm{for } x \ge a

For s \ne 0:

f(x) = \frac{1}{b}\left(1 + s\frac{x-a}{b}\right)^{-1/s - 1}

where 1 + s(x-a)/b > 0

Cumulative distribution function (c.d.f):

For s = 0:

F(x) = 1 - \exp\left(-\frac{x-a}{b}\right) \quad \textrm{for } x \ge a

For s \ne 0:

F(x) = 1 - \left(1 + s\frac{x-a}{b}\right)^{-1/s}

where 1 + s(x-a)/b > 0

Quantile function:

For s = 0:

Q(p) = a - b\log(1-p)

For s \ne 0:

Q(p) = a + \frac{b}{s}\left[(1-p)^{-s} - 1\right]

Median:

For s = 0:

\textrm{Median}(X) = a + b\log(2)

For s \ne 0:

\textrm{Median}(X) = a + \frac{b}{s}\left(2^s - 1\right)

Skewness and Kurtosis: No closed-form expressions; approximated numerically.

See Also

evd::dgpd()

Examples

dist <- dist_gpd(location = 0, scale = 1, shape = 0)

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Gumbel distribution

Description

The Gumbel distribution is a special case of the Generalized Extreme Value 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

dist_gumbel(alpha, scale)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_gumbel.html

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

Support: R, the set of all real numbers.

Mean:

E(X) = \alpha + \sigma\gamma

where \gamma is the Euler-Mascheroni constant, approximately equal to 0.5772157.

Variance:

\textrm{Var}(X) = \frac{\pi^2 \sigma^2}{6}

Skewness:

\textrm{Skew}(X) = \frac{12\sqrt{6}\zeta(3)}{\pi^3} \approx 1.1395

where \zeta(3) is Apery's constant, approximately equal to 1.2020569. Note that skewness is independent of the distribution parameters.

Kurtosis (excess):

\textrm{Kurt}(X) = \frac{12}{5} = 2.4

Note that excess kurtosis is independent of the distribution parameters.

Median:

\textrm{Median}(X) = \alpha - \sigma\ln(\ln 2)

Probability density function (p.d.f):

f(x) = \frac{1}{\sigma} \exp\left[-\frac{x - \alpha}{\sigma}\right] \exp\left\{-\exp\left[-\frac{x - \alpha}{\sigma}\right]\right\}

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

Cumulative distribution function (c.d.f):

F(x) = \exp\left\{-\exp\left[-\frac{x - \alpha}{\sigma}\right]\right\}

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

Quantile function (inverse c.d.f):

F^{-1}(p) = \alpha - \sigma \ln(-\ln p)

for p in (0, 1).

Moment generating function (m.g.f):

E(e^{tX}) = \Gamma(1 - \sigma t) e^{\alpha t}

for \sigma t < 1, where \Gamma is the gamma function.

See Also

actuar::Gumbel, actuar::dgumbel(), actuar::pgumbel(), actuar::qgumbel(), actuar::rgumbel(), actuar::mgumbel()

Examples

dist <- dist_gumbel(alpha = c(0.5, 1, 1.5, 3), scale = c(2, 2, 3, 4))
dist


mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The 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.

Usage

dist_hypergeometric(m, n, k)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_hypergeometric.html

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}{m+n} = kp

Variance: \frac{kmn(m+n-k)}{(m+n)^2 (m+n-1)} = kp(1-p)\left(1 - \frac{k-1}{m+n-1}\right)

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 x) = \sum_{i = \max(0, k-n)}^{\lfloor x \rfloor} \frac{{m \choose i}{n \choose k-i}}{{m+n \choose k}}

Moment generating function (m.g.f):

E(e^{tX}) = \frac{{m \choose k}}{{m+n \choose k}}{}_2F_1(-m, -k; m+n-k+1; e^t)

where _2F_1 is the hypergeometric function.

Skewness:

\frac{(m+n-2k)(m+n-1)^{1/2}(m+n-2n)}{[kmn(m+n-k)]^{1/2}(m+n-2)}

See Also

stats::Hypergeometric

Examples

dist <- dist_hypergeometric(m = rep(500, 3), n = c(50, 60, 70), k = c(100, 200, 300))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

Inflate a value of a probability distribution

Description

Inflated distributions add extra probability mass at a specific value, most commonly zero (zero-inflation). These distributions are useful for modeling data with excess observations at a particular value compared to what the base distribution would predict. Common applications include zero-inflated Poisson or negative binomial models for count data with many zeros.

Usage

dist_inflated(dist, prob, x = 0)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_inflated.html

In the following, let Y be an inflated random variable based on a base distribution X, with inflation value x = c and inflation probability prob = p.

Support: Same as the base distribution, but with additional probability mass at c

Mean: (when x is numeric)

E(Y) = p \cdot c + (1-p) \cdot E(X)

Variance: (when x = 0)

\text{Var}(Y) = (1-p) \cdot \text{Var}(X) + p(1-p) \cdot [E(X)]^2

For non-zero inflation values, the variance is not computed in closed form.

Probability mass/density function (p.m.f/p.d.f):

For discrete distributions:

f_Y(y) = \begin{cases} p + (1-p) \cdot f_X(c) & \text{if } y = c \\ (1-p) \cdot f_X(y) & \text{if } y \neq c \end{cases}

For continuous distributions:

f_Y(y) = \begin{cases} p & \text{if } y = c \\ (1-p) \cdot f_X(y) & \text{if } y \neq c \end{cases}

Cumulative distribution function (c.d.f):

F_Y(q) = \begin{cases} (1-p) \cdot F_X(q) & \text{if } q < c \\ p + (1-p) \cdot F_X(q) & \text{if } q \geq c \end{cases}

Quantile function:

The quantile function is computed numerically by inverting the inflated CDF, accounting for the jump in probability at the inflation point.

Examples

# Zero-inflated Poisson
dist <- dist_inflated(dist_poisson(lambda = 2), prob = 0.3, x = 0)

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, 0)
density(dist, 1)

cdf(dist, 2)

quantile(dist, 0.5)

The Inverse Exponential distribution

Description

The Inverse Exponential distribution is used to model the reciprocal of exponentially distributed variables.

Usage

dist_inverse_exponential(rate)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_inverse_exponential.html

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

Support: x > 0

Mean: Does not exist, returns NA

Variance: Does not exist, returns NA

Probability density function (p.d.f):

f(x) = \frac{\lambda}{x^2} e^{-\lambda/x}

Cumulative distribution function (c.d.f):

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

Quantile function (inverse c.d.f):

F^{-1}(p) = -\frac{\lambda}{\log(p)}

Moment generating function (m.g.f):

Does not exist (divergent integral).

See Also

actuar::InverseExponential

Examples

dist <- dist_inverse_exponential(rate = 1:5)
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Inverse Gamma distribution

Description

The Inverse Gamma distribution is commonly used as a prior distribution in Bayesian statistics, particularly for variance parameters.

Usage

dist_inverse_gamma(shape, rate = 1/scale, scale)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_inverse_gamma.html

In the following, let X be an Inverse Gamma random variable with shape parameter shape = \alpha and rate parameter rate = \beta (equivalently, scale = 1/\beta).

Support: x \in (0, \infty)

Mean: \frac{\beta}{\alpha - 1} for \alpha > 1, otherwise undefined

Variance: \frac{\beta^2}{(\alpha - 1)^2 (\alpha - 2)} for \alpha > 2, otherwise undefined

Probability density function (p.d.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)} = Q(\alpha, \beta/x)

where \Gamma(\alpha, z) is the upper incomplete gamma function and Q is the regularized incomplete gamma function.

Moment generating function (m.g.f):

M_X(t) = \frac{2 (-\beta t)^{\alpha/2}}{\Gamma(\alpha)} K_\alpha\left(\sqrt{-4\beta t}\right)

for t < 0, where K_\alpha is the modified Bessel function of the second kind. The MGF does not exist for t \ge 0.

See Also

actuar::InverseGamma

Examples

dist <- dist_inverse_gamma(shape = c(1,2,3,3), rate = c(1,1,1,2))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Inverse Gaussian distribution

Description

Usage

dist_inverse_gaussian(mean, shape)

Arguments

Details

The inverse Gaussian distribution (also known as the Wald distribution) is commonly used to model positive-valued data, particularly in contexts involving first passage times and reliability analysis.

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_inverse_gaussian.html

In the following, let X be an Inverse Gaussian random variable with parameters mean = \mu and shape = \lambda.

Support: (0, \infty)

Mean: \mu

Variance: \frac{\mu^3}{\lambda}

Probability density function (p.d.f):

f(x) = \sqrt{\frac{\lambda}{2\pi x^3}} \exp\left(-\frac{\lambda(x - \mu)^2}{2\mu^2 x}\right)

Cumulative distribution function (c.d.f):

F(x) = \Phi\left(\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu} - 1\right)\right) + \exp\left(\frac{2\lambda}{\mu}\right) \Phi\left(-\sqrt{\frac{\lambda}{x}} \left(\frac{x}{\mu} + 1\right)\right)

where \Phi is the standard normal c.d.f.

Moment generating function (m.g.f):

E(e^{tX}) = \exp\left(\frac{\lambda}{\mu} \left(1 - \sqrt{1 - \frac{2\mu^2 t}{\lambda}}\right)\right)

for t < \frac{\lambda}{2\mu^2}.

Skewness: 3\sqrt{\frac{\mu}{\lambda}}

Excess Kurtosis: \frac{15\mu}{\lambda}

Quantiles: No closed-form expression, approximated numerically.

See Also

actuar::InverseGaussian

Examples

dist <- dist_inverse_gaussian(mean = c(1,1,1,3,3), shape = c(0.2, 1, 3, 0.2, 1))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Laplace distribution

Description

The Laplace distribution, also known as the double exponential distribution, is a continuous probability distribution that is symmetric around its location parameter.

Usage

dist_laplace(mu, sigma)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_laplace.html

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

Support: R, the set of all real numbers

Mean: \mu

Variance: 2\sigma^2

Probability density function (p.d.f):

f(x) = \frac{1}{2\sigma} \exp\left(-\frac{|x - \mu|}{\sigma}\right)

Cumulative distribution function (c.d.f):

F(x) = \begin{cases} \frac{1}{2} \exp\left(\frac{x - \mu}{\sigma}\right) & \text{if } x < \mu \\ 1 - \frac{1}{2} \exp\left(-\frac{x - \mu}{\sigma}\right) & \text{if } x \geq \mu \end{cases}

Moment generating function (m.g.f):

E(e^{tX}) = \frac{\exp(\mu t)}{1 - \sigma^2 t^2} \text{ for } |t| < \frac{1}{\sigma}

See Also

extraDistr::Laplace

Examples

dist <- dist_laplace(mu = c(0, 2, -1), sigma = c(1, 2, 0.5))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 0)
density(dist, 0, log = TRUE)

cdf(dist, 1)

quantile(dist, 0.7)

The Logarithmic distribution

Description

The Logarithmic distribution is a discrete probability distribution derived from the logarithmic series. It is useful in modeling the abundance of species and other phenomena where the frequency of an event follows a logarithmic pattern.

Usage

dist_logarithmic(prob)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_logarithmic.html

In the following, let X be a Logarithmic random variable with parameter prob = p.

Support: \{1, 2, 3, ...\}

Mean: \frac{-1}{\log(1-p)} \cdot \frac{p}{1-p}

Variance: \frac{-(p^2 + p\log(1-p))}{[(1-p)\log(1-p)]^2}

Probability mass function (p.m.f):

P(X = k) = \frac{-1}{\log(1-p)} \cdot \frac{p^k}{k}

for k = 1, 2, 3, \ldots

Cumulative distribution function (c.d.f):

The c.d.f. does not have a simple closed form. It is computed using the recurrence relationship P(X = k+1) = \frac{p \cdot k}{k+1} \cdot P(X = k) starting from P(X = 1) = \frac{-p}{\log(1-p)}.

Moment generating function (m.g.f):

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

for pe^t < 1

See Also

actuar::Logarithmic

Examples

dist <- dist_logarithmic(prob = c(0.33, 0.66, 0.99))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The 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

dist_logistic(location, scale)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_logistic.html

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 \left[1 + e^{-\frac{x - \mu}{s}}\right]^2}

Cumulative distribution function (c.d.f):

F(x) = \frac{1}{1 + e^{-\frac{x - \mu}{s}}}

Moment generating function (m.g.f):

E(e^{tX}) = e^{\mu t} B(1 - st, 1 + st)

for -1 < st < 1, where B(a, b) is the Beta function.

See Also

stats::Logistic

Examples

dist <- dist_logistic(location = c(5,9,9,6,2), scale = c(2,3,4,2,1))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The log-normal distribution

Description

The log-normal distribution is a commonly used transformation of the Normal distribution. If X follows a log-normal distribution, then \ln{X} would be characterised by a Normal distribution.

Usage

dist_lognormal(mu = 0, sigma = 1)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_lognormal.html

In the following, let X be a log-normal random variable with mu = \mu and sigma = \sigma.

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

Mean: e^{\mu + \sigma^2/2}

Variance: (e^{\sigma^2} - 1) e^{2\mu + \sigma^2}

Skewness: (e^{\sigma^2} + 2) \sqrt{e^{\sigma^2} - 1}

Excess Kurtosis: e^{4\sigma^2} + 2 e^{3\sigma^2} + 3 e^{2\sigma^2} - 6

Probability density function (p.d.f):

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

Cumulative distribution function (c.d.f):

F(x) = \Phi\left(\frac{\ln{x} - \mu}{\sigma}\right)

where \Phi is the c.d.f. of the standard Normal distribution.

Moment generating function (m.g.f):

Does not exist in closed form.

See Also

stats::Lognormal

Examples

dist <- dist_lognormal(mu = 1:5, sigma = 0.1)

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

# A log-normal distribution X is exp(Y), where Y is a Normal distribution of
# the same parameters. So log(X) will produce the Normal distribution Y.
log(dist)

Missing distribution

Description

A placeholder distribution for handling missing values in a vector of distributions.

Usage

dist_missing(length = 1)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_missing.html

The missing distribution represents the absence of distributional information. It is used as a placeholder when distribution values are not available or not applicable, similar to how NA is used for missing scalar values.

Support: Undefined

Mean: \text{NA}

Variance: \text{NA}

Skewness: \text{NA}

Kurtosis: \text{NA}

Probability density function (p.d.f): Undefined

f(x) = \text{NA}

Cumulative distribution function (c.d.f): Undefined

F(t) = \text{NA}

Quantile function: Undefined

Q(p) = \text{NA}

Moment generating function (m.g.f): Undefined

E(e^{tX}) = \text{NA}

All statistical operations on missing distributions return NA values of appropriate length, propagating the missingness through calculations.

See Also

base::NA

Examples

dist <- dist_missing(3L)

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

Create a mixture of distributions

Description

A mixture distribution combines multiple component distributions with specified weights. The resulting distribution can model complex, multimodal data by representing it as a weighted sum of simpler distributions.

Usage

dist_mixture(..., weights = numeric())

Arguments

Details

In the following, let X be a mixture random variable composed of K component distributions F_1, F_2, \ldots, F_K with corresponding weights w_1, w_2, \ldots, w_K where \sum_{i=1}^K w_i = 1 and w_i \geq 0 for all i.

Support: The union of the supports of all component distributions

Mean:

For univariate mixtures:

E(X) = \sum_{i=1}^K w_i \mu_i

where \mu_i is the mean of the i-th component distribution.

For multivariate mixtures:

E(\mathbf{X}) = \sum_{i=1}^K w_i \boldsymbol{\mu}_i

where \boldsymbol{\mu}_i is the mean vector of the i-th component distribution.

Variance:

For univariate mixtures:

\text{Var}(X) = \sum_{i=1}^K w_i (\mu_i^2 + \sigma_i^2) - \left(\sum_{i=1}^K w_i \mu_i\right)^2

where \sigma_i^2 is the variance of the i-th component distribution.

Covariance:

For multivariate mixtures:

\text{Cov}(\mathbf{X}) = \sum_{i=1}^K w_i \left[ (\boldsymbol{\mu}_i - \bar{\boldsymbol{\mu}})(\boldsymbol{\mu}_i - \bar{\boldsymbol{\mu}})^T + \boldsymbol{\Sigma}_i \right]

where \bar{\boldsymbol{\mu}} = \sum_{i=1}^K w_i \boldsymbol{\mu}_i is the overall mean vector and \boldsymbol{\Sigma}_i is the covariance matrix of the i-th component distribution.

Probability density/mass function (p.d.f/p.m.f):

f(x) = \sum_{i=1}^K w_i f_i(x)

where f_i(x) is the density or mass function of the i-th component distribution.

Cumulative distribution function (c.d.f):

For univariate mixtures:

F(x) = \sum_{i=1}^K w_i F_i(x)

where F_i(x) is the c.d.f. of the i-th component distribution.

For multivariate mixtures, the c.d.f. is approximated numerically.

Quantile function:

For univariate mixtures, the quantile function has no closed form and is computed numerically by inverting the c.d.f. using root-finding (stats::uniroot()).

For multivariate mixtures, quantiles are not yet implemented.

See Also

stats::uniroot(), vctrs::vec_unique_count()

Examples

# Univariate mixture of two normal distributions
dist <- dist_mixture(dist_normal(0, 1), dist_normal(5, 2), weights = c(0.3, 0.7))
dist

mean(dist)
variance(dist)

density(dist, 2)
cdf(dist, 2)
quantile(dist, 0.5)

generate(dist, 10)

The 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 dist_bernoulli() distribution to include more than two categories, resulting in a dist_categorical() distribution. We then extend repeat the Categorical experiment several (n) times.

Usage

dist_multinomial(size, prob)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multinomial.html

In the following, let X = (X_1, ..., X_k) be a Multinomial random variable with success probability prob = 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! \cdots x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_k^{x_k}

where \sum_{i=1}^k x_i = n and \sum_{i=1}^k p_i = 1.

Cumulative distribution function (c.d.f):

P(X_1 \le q_1, ..., X_k \le q_k) = \sum_{\substack{x_1, \ldots, x_k \ge 0 \\ x_i \le q_i \text{ for all } i \\ \sum_{i=1}^k x_i = n}} \frac{n!}{x_1! x_2! \cdots x_k!} p_1^{x_1} \cdot p_2^{x_2} \cdot \ldots \cdot p_k^{x_k}

The c.d.f. is computed as a finite sum of the p.m.f. over all integer vectors in the support that satisfy the componentwise inequalities.

Moment generating function (m.g.f):

E(e^{t'X}) = \left(\sum_{i=1}^k p_i e^{t_i}\right)^n

where t = (t_1, ..., t_k) is a vector of the same dimension as X.

Skewness: The skewness of X_i is

\frac{1 - 2p_i}{\sqrt{n p_i (1 - p_i)}}

Excess Kurtosis: The excess kurtosis of X_i is

\frac{1 - 6p_i(1 - p_i)}{n p_i (1 - p_i)}

See Also

stats::dmultinom(), stats::rmultinom()

Examples

dist <- dist_multinomial(size = c(4, 3), prob = list(c(0.3, 0.5, 0.2), c(0.1, 0.5, 0.4)))

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, list(d = rbind(cbind(1,2,1), cbind(0,2,1))))
density(dist, list(d = rbind(cbind(1,2,1), cbind(0,2,1))), log = TRUE)

cdf(dist, cbind(1,2,1))

The multivariate normal distribution

Description

The multivariate normal distribution is a generalization of the univariate normal distribution to higher dimensions. It is widely used in multivariate statistics and describes the joint distribution of multiple correlated continuous random variables.

Usage

dist_multivariate_normal(mu = 0, sigma = diag(1))

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multivariate_normal.html

In the following, let \mathbf{X} be a k-dimensional multivariate normal random variable with mean vector mu = \boldsymbol{\mu} and variance-covariance matrix sigma = \boldsymbol{\Sigma}.

Support: \mathbf{x} \in \mathbb{R}^k

Mean: \boldsymbol{\mu}

Variance-covariance matrix: \boldsymbol{\Sigma}

Probability density function (p.d.f):

f(\mathbf{x}) = \frac{1}{(2\pi)^{k/2} |\boldsymbol{\Sigma}|^{1/2}} \exp\left(-\frac{1}{2}(\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1}(\mathbf{x} - \boldsymbol{\mu})\right)

where |\boldsymbol{\Sigma}| is the determinant of \boldsymbol{\Sigma}.

Cumulative distribution function (c.d.f):

P(\mathbf{X} \le \mathbf{q}) = P(X_1 \le q_1, \ldots, X_k \le q_k)

The c.d.f. does not have a closed-form expression and is computed numerically.

Moment generating function (m.g.f):

M(\mathbf{t}) = E(e^{\mathbf{t}^T \mathbf{X}}) = \exp\left(\mathbf{t}^T \boldsymbol{\mu} + \frac{1}{2}\mathbf{t}^T \boldsymbol{\Sigma} \mathbf{t}\right)

See Also

mvtnorm::dmvnorm(), mvtnorm::pmvnorm(), mvtnorm::qmvnorm(), mvtnorm::rmvnorm()

Examples

dist <- dist_multivariate_normal(mu = list(c(1,2)), sigma = list(matrix(c(4,2,2,3), ncol=2)))
dimnames(dist) <- c("x", "y")
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, cbind(2, 1))
density(dist, cbind(2, 1), log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7, kind = "equicoordinate")
quantile(dist, 0.7, kind = "marginal")

The multivariate t-distribution

Description

The multivariate t-distribution is a generalization of the univariate Student's t-distribution to multiple dimensions. It is commonly used for modeling heavy-tailed multivariate data and in robust statistics.

Usage

dist_multivariate_t(df = 1, mu = 0, sigma = diag(1))

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_multivariate_t.html

In the following, let \mathbf{X} be a multivariate t random vector with degrees of freedom df = \nu, location parameter mu = \boldsymbol{\mu}, and scale matrix sigma = \boldsymbol{\Sigma}.

Support: \mathbf{x} \in \mathbb{R}^k, where k is the dimension of the distribution

Mean: \boldsymbol{\mu} for \nu > 1, undefined otherwise

Covariance matrix:

\text{Cov}(\mathbf{X}) = \frac{\nu}{\nu - 2} \boldsymbol{\Sigma}

for \nu > 2, undefined otherwise

Probability density function (p.d.f):

f(\mathbf{x}) = \frac{\Gamma\left(\frac{\nu + k}{2}\right)} {\Gamma\left(\frac{\nu}{2}\right) \nu^{k/2} \pi^{k/2} |\boldsymbol{\Sigma}|^{1/2}} \left[1 + \frac{1}{\nu}(\mathbf{x} - \boldsymbol{\mu})^T \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu})\right]^{-\frac{\nu + k}{2}}

where k is the dimension of the distribution and \Gamma(\cdot) is the gamma function.

Cumulative distribution function (c.d.f):

F(\mathbf{t}) = \int_{-\infty}^{t_1} \cdots \int_{-\infty}^{t_k} f(\mathbf{x}) \, d\mathbf{x}

This integral does not have a closed form solution and is approximated numerically.

Quantile function:

The equicoordinate quantile function finds q such that:

P(X_1 \leq q, \ldots, X_k \leq q) = p

This does not have a closed form solution and is approximated numerically.

The marginal quantile function for each dimension i is:

Q_i(p) = \mu_i + \sqrt{\Sigma_{ii}} \cdot t_{\nu}^{-1}(p)

where t_{\nu}^{-1}(p) is the quantile function of the univariate Student's t-distribution with \nu degrees of freedom, and \Sigma_{ii} is the i-th diagonal element of sigma.

See Also

mvtnorm::dmvt, mvtnorm::pmvt, mvtnorm::qmvt, mvtnorm::rmvt

Examples

dist <- dist_multivariate_t(
  df = 5,
  mu = list(c(1, 2)),
  sigma = list(matrix(c(4, 2, 2, 3), ncol = 2))
)
dimnames(dist) <- c("x", "y")
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, cbind(2, 1))
density(dist, cbind(2, 1), log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)
quantile(dist, 0.7, kind = "marginal")

The 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 number of successes (size) occur. The probability of success in each trial is given by prob.

Usage

dist_negative_binomial(size, prob)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_negative_binomial.html

In the following, let X be a Negative Binomial random variable with success probability prob = p and the number of successes size = r.

Support: \{0, 1, 2, 3, ...\}

Mean: \frac{r(1-p)}{p}

Variance: \frac{r(1-p)}{p^2}

Probability mass function (p.m.f):

P(X = k) = \binom{k + r - 1}{k} (1-p)^r p^k

Cumulative distribution function (c.d.f):

F(k) = \sum_{i=0}^{\lfloor k \rfloor} \binom{i + r - 1}{i} (1-p)^r p^i

This can also be expressed in terms of the regularized incomplete beta function, and is computed numerically.

Moment generating function (m.g.f):

E(e^{tX}) = \left(\frac{1-p}{1-pe^t}\right)^r, \quad t < -\log p

Skewness:

\gamma_1 = \frac{2-p}{\sqrt{r(1-p)}}

Excess Kurtosis:

\gamma_2 = \frac{6}{r} + \frac{p^2}{r(1-p)}

See Also

stats::NegBinomial

Examples

dist <- dist_negative_binomial(size = 10, prob = 0.5)

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)
support(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The 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

dist_normal(mu = 0, sigma = 1, mean = mu, sd = sigma)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_normal.html

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):

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

This integral does not have a closed form solution and is 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}

See Also

stats::Normal

Examples

dist <- dist_normal(mu = 1:5, sigma = 3)

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Pareto Distribution

Description

The Pareto distribution is a power-law probability distribution commonly used in actuarial science to model loss severity and in economics to model income distributions and firm sizes.

Usage

dist_pareto(shape, scale)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_pareto.html

In the following, let X be a Pareto random variable with parameters shape = \alpha and scale = \theta.

Support: (0, \infty)

Mean: \frac{\theta}{\alpha - 1} for \alpha > 1, undefined otherwise

Variance: \frac{\alpha\theta^2}{(\alpha - 1)^2(\alpha - 2)} for \alpha > 2, undefined otherwise

Probability density function (p.d.f):

f(x) = \frac{\alpha\theta^\alpha}{(x + \theta)^{\alpha + 1}}

for x > 0, \alpha > 0 and \theta > 0.

Cumulative distribution function (c.d.f):

F(x) = 1 - \left(\frac{\theta}{x + \theta}\right)^\alpha

for x > 0.

Moment generating function (m.g.f):

Does not exist in closed form, but the kth raw moment E[X^k] exists for -1 < k < \alpha.

Note

There are many different definitions of the Pareto distribution in the literature; see Arnold (2015) or Kleiber and Kotz (2003). This implementation uses the Pareto distribution without a location parameter as described in actuar::Pareto.

References

Kleiber, C. and Kotz, S. (2003), Statistical Size Distributions in Economics and Actuarial Sciences, Wiley.

Klugman, S. A., Panjer, H. H. and Willmot, G. E. (2012), Loss Models, From Data to Decisions, Fourth Edition, Wiley.

See Also

actuar::Pareto

Examples

dist <- dist_pareto(shape = c(10, 3, 2, 1), scale = rep(1, 4))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

Percentile distribution

Description

The Percentile distribution is a non-parametric distribution defined by a set of quantiles at specified percentile values. This distribution is useful for representing empirical distributions or elicited expert knowledge when only percentile information is available. The distribution uses linear interpolation between percentiles and can be used to approximate complex distributions that may not have simple parametric forms.

Usage

dist_percentile(x, percentile)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_percentile.html

In the following, let X be a Percentile random variable defined by values x_1, x_2, \ldots, x_n at percentiles p_1, p_2, \ldots, p_n where 0 \le p_i \le 100.

Support: [\min(x_i), \max(x_i)] if \min(p_i) > 0 or \max(p_i) < 100, otherwise support is approximated from the specified percentiles.

Mean: Approximated numerically using spline interpolation and numerical integration:

E(X) \approx \int_0^1 Q(u) du

where Q(u) is a spline function interpolating the percentile values.

Variance: Approximated numerically.

Probability density function (p.d.f): Approximated numerically using kernel density estimation from generated samples.

Cumulative distribution function (c.d.f): Defined by linear interpolation:

F(t) = \begin{cases} p_1/100 & \text{if } t < x_1 \\ p_i/100 + \frac{(t - x_i)(p_{i+1} - p_i)}{100(x_{i+1} - x_i)} & \text{if } x_i \le t < x_{i+1} \\ p_n/100 & \text{if } t \ge x_n \end{cases}

Quantile function: Defined by linear interpolation:

Q(u) = x_i + \frac{(100u - p_i)(x_{i+1} - x_i)}{p_{i+1} - p_i}

for p_i/100 \le u \le p_{i+1}/100.

Examples

dist <- dist_normal()
percentiles <- seq(0.01, 0.99, by = 0.01)
x <- vapply(percentiles, quantile, double(1L), x = dist)
dist_percentile(list(x), list(percentiles*100))

The Poisson Distribution

Description

Poisson distributions are frequently used to model counts. The Poisson distribution is commonly used to model the number of events occurring in a fixed interval of time or space when these events occur with a known constant mean rate and independently of the time since the last event. Examples include the number of emails received per hour, the number of decay events per second from a radioactive source, or the number of customers arriving at a store per day.

Usage

dist_poisson(lambda)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_poisson.html

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)}

Skewness:

\gamma_1 = \frac{1}{\sqrt{\lambda}}

Excess kurtosis:

\gamma_2 = \frac{1}{\lambda}

See Also

stats::Poisson

Examples

dist <- dist_poisson(lambda = c(1, 4, 10))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Poisson-Inverse Gaussian distribution

Description

The Poisson-Inverse Gaussian distribution is a compound Poisson distribution where the rate parameter follows an Inverse Gaussian distribution. It is useful for modeling overdispersed count data.

Usage

dist_poisson_inverse_gaussian(mean, shape)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_poisson_inverse_gaussian.html

In the following, let X be a Poisson-Inverse Gaussian random variable with parameters mean = \mu and shape = \phi.

Support: \{0, 1, 2, 3, ...\}

Mean: \mu

Variance: \frac{\mu}{\phi}(\mu^2 + \phi)

Probability mass function (p.m.f):

P(X = x) = \frac{e^{\phi}}{\sqrt{2\pi}} \left(\frac{\phi}{\mu^2}\right)^{x/2} \frac{1}{x!} \int_0^\infty u^{x-1/2} \exp\left(-\frac{\phi u}{2} - \frac{\phi}{2\mu^2 u}\right) du

for x = 0, 1, 2, \ldots

Cumulative distribution function (c.d.f):

P(X \le x) = \sum_{k=0}^{\lfloor x \rfloor} P(X = k)

The c.d.f does not have a closed form and is approximated numerically.

Moment generating function (m.g.f):

E(e^{tX}) = \exp\left\{\phi\left[1 - \sqrt{1 - \frac{2\mu^2}{\phi}(e^t - 1)}\right]\right\}

for t < -\log(1 + \phi/(2\mu^2))

See Also

actuar::PoissonInverseGaussian, actuar::dpoisinvgauss(), actuar::ppoisinvgauss(), actuar::qpoisinvgauss(), actuar::rpoisinvgauss()

Examples

dist <- dist_poisson_inverse_gaussian(mean = rep(0.1, 3), shape = c(0.4, 0.8, 1))
dist


mean(dist)
variance(dist)
support(dist)
generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

Sampling distribution

Description

The sampling distribution represents an empirical distribution based on observed samples. It is useful for bootstrapping, representing posterior distributions from Markov Chain Monte Carlo (MCMC) algorithms, or working with any empirical data where the parametric form is unknown. Unlike parametric distributions, the sampling distribution makes no assumptions about the underlying data-generating process and instead uses the sample itself to estimate distributional properties. The distribution can handle both univariate and multivariate samples.

Usage

dist_sample(x)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_sample.html

In the following, let X be a random variable with sample x_1, x_2, \ldots, x_n of size n.

Support: The observed range of the sample

Mean (univariate):

\bar{x} = \frac{1}{n} \sum_{i=1}^{n} x_i

Mean (multivariate): Computed independently for each variable.

Variance (univariate):

s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2

Covariance (multivariate): The sample covariance matrix.

Skewness (univariate):

g_1 = \frac{\sqrt{n} \sum_{i=1}^{n} (x_i - \bar{x})^3}{\left(\sum_{i=1}^{n} (x_i - \bar{x})^2\right)^{3/2}} \left(1 - \frac{1}{n}\right)^{3/2}

Probability density function: Approximated numerically using kernel density estimation.

Cumulative distribution function (univariate):

F(q) = \frac{1}{n} \sum_{i=1}^{n} I(x_i \leq q)

where I(\cdot) is the indicator function.

Cumulative distribution function (multivariate):

F(\mathbf{q}) = \frac{1}{n} \sum_{i=1}^{n} I(\mathbf{x}_i \leq \mathbf{q})

where the inequality is applied element-wise.

Quantile function (univariate): The sample quantile, computed using the specified quantile type (see stats::quantile()).

Quantile function (multivariate): Marginal quantiles are computed independently for each variable.

Random generation: Bootstrap sampling with replacement from the empirical sample.

See Also

stats::density(), stats::quantile(), stats::cov()

Examples

# Univariate numeric samples
dist <- dist_sample(x = list(rnorm(100), rnorm(100, 10)))

dist
mean(dist)
variance(dist)
skewness(dist)
generate(dist, 10)

density(dist, 1)

# Multivariate numeric samples
dist <- dist_sample(x = list(cbind(rnorm(100), rnorm(100, 10))))
dimnames(dist) <- c("x", "y")

dist
mean(dist)
variance(dist)
generate(dist, 10)
quantile(dist, 0.4) # Returns the marginal quantiles
cdf(dist, matrix(c(0.3,9), nrow = 1))

The (non-central) location-scale Student 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

dist_student_t(df, mu = 0, sigma = 1, ncp = NULL)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_student_t.html

In the following, let X be a location-scale Student's T random variable with df = \nu, mu = \mu, sigma = \sigma, and ncp = \delta (non-centrality parameter).

If Z follows a standard Student's T distribution (with df = \nu and ncp = \delta), then X = \mu + \sigma Z.

Support: R, the set of all real numbers

Mean:

For the central distribution (ncp = 0 or NULL):

E(X) = \mu

for \nu > 1, and undefined otherwise.

For the non-central distribution (ncp \neq 0):

E(X) = \mu + \delta \sqrt{\frac{\nu}{2}} \frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)} \sigma

for \nu > 1, and undefined otherwise.

Variance:

For the central distribution (ncp = 0 or NULL):

\mathrm{Var}(X) = \frac{\nu}{\nu - 2} \sigma^2

for \nu > 2. Undefined if \nu \le 1, infinite when 1 < \nu \le 2.

For the non-central distribution (ncp \neq 0):

\mathrm{Var}(X) = \left[\frac{\nu(1+\delta^2)}{\nu-2} - \left(\delta \sqrt{\frac{\nu}{2}} \frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)}\right)^2\right] \sigma^2

for \nu > 2. Undefined if \nu \le 1, infinite when 1 < \nu \le 2.

Probability density function (p.d.f):

For the central distribution (ncp = 0 or NULL), the standard t distribution with df = \nu has density:

f_Z(z) = \frac{\Gamma((\nu + 1)/2)}{\sqrt{\pi \nu} \Gamma(\nu/2)} \left(1 + \frac{z^2}{\nu} \right)^{- (\nu + 1)/2}

The location-scale version with mu = \mu and sigma = \sigma has density:

f(x) = \frac{1}{\sigma} f_Z\left(\frac{x - \mu}{\sigma}\right)

For the non-central distribution (ncp \neq 0), the density is computed numerically via stats::dt().

Cumulative distribution function (c.d.f):

For the central distribution (ncp = 0 or NULL), the cumulative distribution function is computed numerically via stats::pt(), which uses the relationship to the incomplete beta function:

F_\nu(t) = \frac{1}{2} I_x\left(\frac{\nu}{2}, \frac{1}{2}\right)

for t \le 0, where x = \nu/(\nu + t^2) and I_x(a,b) is the incomplete beta function (stats::pbeta()). For t \ge 0:

F_\nu(t) = 1 - \frac{1}{2} I_x\left(\frac{\nu}{2}, \frac{1}{2}\right)

The location-scale version is: F(x) = F_\nu((x - \mu)/\sigma).

For the non-central distribution (ncp \neq 0), the cumulative distribution function is computed numerically via stats::pt().

Moment generating function (m.g.f):

Does not exist in closed form. Moments are computed using the formulas for mean and variance above where available.

See Also

stats::TDist

Examples

dist <- dist_student_t(df = c(1,2,5), mu = c(0,1,2), sigma = c(1,2,3))

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Studentized Range distribution

Description

Tukey's studentized range distribution, used for Tukey's honestly significant differences test in ANOVA.

Usage

dist_studentized_range(nmeans, df, nranges)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_studentized_range.html

In the following, let Q be a Studentized Range random variable with parameters nmeans = k (number of groups), df = \nu (degrees of freedom), and nranges = n (number of ranges).

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

Mean: Approximated numerically.

Variance: Approximated numerically.

Probability density function (p.d.f): The density does not have a closed-form expression and is computed numerically.

Cumulative distribution function (c.d.f): The c.d.f does not have a simple closed-form expression. For n = 1 (single range), it involves integration over the joint distribution of the sample range and an independent chi-square variable. The general form is computed numerically using algorithms described in the references for stats::ptukey().

Moment generating function (m.g.f): Does not exist in closed form.

See Also

stats::Tukey

Examples

dist <- dist_studentized_range(nmeans = c(6, 2), df = c(5, 4), nranges = c(1, 1))

dist

cdf(dist, 4)

quantile(dist, 0.7)

Modify a distribution with a transformation

Description

A transformed distribution applies a monotonic transformation to an existing distribution. This is useful for creating derived distributions such as log-normal (exponential transformation of normal), or other custom transformations of base distributions.

The density(), mean(), and variance() methods are approximate as they are based on numerical derivatives.

Usage

dist_transformed(dist, transform, inverse)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_transformed.html

Let Y = g(X) where X is the base distribution with transformation function transform = g and inverse = g^{-1}. The transformation g must be monotonic over the support of X.

Support: g(S_X) where S_X is the support of X

Mean: Approximated numerically using a second-order Taylor expansion:

E(Y) \approx g(\mu_X) + \frac{1}{2}g''(\mu_X)\sigma_X^2

where \mu_X and \sigma_X^2 are the mean and variance of the base distribution X, and g'' is the second derivative of the transformation. The derivative is computed numerically using numDeriv::hessian().

Variance: Approximated numerically using the delta method:

\mathrm{Var}(Y) \approx [g'(\mu_X)]^2\sigma_X^2 + \frac{1}{2}[g''(\mu_X)\sigma_X^2]^2

where g' is the first derivative (Jacobian) computed numerically using numDeriv::jacobian().

Probability density function (p.d.f): Using the change of variables formula:

f_Y(y) = f_X(g^{-1}(y)) \left|\frac{d}{dy}g^{-1}(y)\right|

where f_X is the p.d.f. of the base distribution and the Jacobian |d/dy \, g^{-1}(y)| is computed numerically using numDeriv::jacobian().

Cumulative distribution function (c.d.f):

For monotonically increasing g:

F_Y(y) = F_X(g^{-1}(y))

For monotonically decreasing g:

F_Y(y) = 1 - F_X(g^{-1}(y))

where F_X is the c.d.f. of the base distribution.

Quantile function: The inverse of the c.d.f.

For monotonically increasing g:

Q_Y(p) = g(Q_X(p))

For monotonically decreasing g:

Q_Y(p) = g(Q_X(1-p))

where Q_X is the quantile function of the base distribution.

See Also

numDeriv::jacobian(), numDeriv::hessian()

Examples

# Create a log normal distribution
dist <- dist_transformed(dist_normal(0, 0.5), exp, log)
density(dist, 1) # dlnorm(1, 0, 0.5)
cdf(dist, 4) # plnorm(4, 0, 0.5)
quantile(dist, 0.1) # qlnorm(0.1, 0, 0.5)
generate(dist, 10) # rlnorm(10, 0, 0.5)

Truncate a distribution

Description

Note that the samples are generated using inverse transform sampling, and the means and variances are estimated from samples.

Usage

dist_truncated(dist, lower = -Inf, upper = Inf)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_truncated.html

In the following, let X be a truncated random variable with underlying distribution Y, truncation bounds lower = a and upper = b, where F_Y(x) is the c.d.f. of Y and f_Y(x) is the p.d.f. of Y.

Support: [a, b]

Mean: For the general case, the mean is approximated numerically. For a truncated Normal distribution with underlying mean \mu and standard deviation \sigma, the mean is:

E(X) = \mu + \frac{\phi(\alpha) - \phi(\beta)}{\Phi(\beta) - \Phi(\alpha)} \sigma

where \alpha = (a - \mu)/\sigma, \beta = (b - \mu)/\sigma, \phi is the standard Normal p.d.f., and \Phi is the standard Normal c.d.f.

Variance: Approximated numerically for all distributions.

Probability density function (p.d.f):

f(x) = \begin{cases} \frac{f_Y(x)}{F_Y(b) - F_Y(a)} & \text{if } a \le x \le b \\ 0 & \text{otherwise} \end{cases}

Cumulative distribution function (c.d.f):

F(x) = \begin{cases} 0 & \text{if } x < a \\ \frac{F_Y(x) - F_Y(a)}{F_Y(b) - F_Y(a)} & \text{if } a \le x \le b \\ 1 & \text{if } x > b \end{cases}

Quantile function:

Q(p) = F_Y^{-1}(F_Y(a) + p(F_Y(b) - F_Y(a)))

clamped to the interval [a, b].

Examples

dist <- dist_truncated(dist_normal(2,1), lower = 0)

dist
mean(dist)
variance(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

if(requireNamespace("ggdist")) {
library(ggplot2)
ggplot() +
  ggdist::stat_dist_halfeye(
    aes(y = c("Normal", "Truncated"),
        dist = c(dist_normal(2,1), dist_truncated(dist_normal(2,1), lower = 0)))
  )
}

The Uniform distribution

Description

A distribution with constant density on an interval.

Usage

dist_uniform(min, max)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_uniform.html

In the following, let X be a Uniform random variable with parameters min = a and max = b.

Support: [a, b]

Mean: \frac{a + b}{2}

Variance: \frac{(b - a)^2}{12}

Probability density function (p.d.f):

f(x) = \frac{1}{b - a}

for x \in [a, b], and f(x) = 0 otherwise.

Cumulative distribution function (c.d.f):

F(x) = \frac{x - a}{b - a}

for x \in [a, b], with F(x) = 0 for x < a and F(x) = 1 for x > b.

Moment generating function (m.g.f):

E(e^{tX}) = \frac{e^{tb} - e^{ta}}{t(b - a)}

for t \neq 0, and E(e^{tX}) = 1 for t = 0.

Skewness: 0

Excess Kurtosis: -\frac{6}{5}

See Also

stats::Uniform

Examples

dist <- dist_uniform(min = c(3, -2), max = c(5, 4))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

The Weibull distribution

Description

Generalization of the gamma distribution. Often used in survival and time-to-event analyses.

Usage

dist_weibull(shape, scale)

Arguments

Details

We recommend reading this documentation on pkgdown which renders math nicely. https://pkg.mitchelloharawild.com/distributional/reference/dist_weibull.html

In the following, let X be a Weibull random variable with shape parameter shape = k and scale parameter scale = \lambda.

Support: [0, \infty)

Mean:

E(X) = \lambda \Gamma\left(1 + \frac{1}{k}\right)

where \Gamma is the gamma function.

Variance:

\text{Var}(X) = \lambda^2 \left[\Gamma\left(1 + \frac{2}{k}\right) - \left(\Gamma\left(1 + \frac{1}{k}\right)\right)^2\right]

Probability density function (p.d.f):

f(x) = \frac{k}{\lambda}\left(\frac{x}{\lambda}\right)^{k-1}e^{-(x/\lambda)^k}, \quad x \ge 0

Cumulative distribution function (c.d.f):

F(x) = 1 - e^{-(x/\lambda)^k}, \quad x \ge 0

Moment generating function (m.g.f):

E(e^{tX}) = \sum_{n=0}^\infty \frac{t^n\lambda^n}{n!} \Gamma\left(1+\frac{n}{k}\right)

Skewness:

\gamma_1 = \frac{\mu^3 - 3\mu\sigma^2 - \mu^3}{\sigma^3}

where \mu = E(X), \sigma^2 = \text{Var}(X), and the third raw moment is

\mu^3 = \lambda^3 \Gamma\left(1 + \frac{3}{k}\right)

Excess Kurtosis:

\gamma_2 = \frac{\mu^4 - 4\gamma_1\mu\sigma^3 - 6\mu^2\sigma^2 - \mu^4}{\sigma^4} - 3

where the fourth raw moment is

\mu^4 = \lambda^4 \Gamma\left(1 + \frac{4}{k}\right)

See Also

stats::Weibull

Examples

dist <- dist_weibull(shape = c(0.5, 1, 1.5, 5), scale = rep(1, 4))

dist
mean(dist)
variance(dist)
skewness(dist)
kurtosis(dist)

generate(dist, 10)

density(dist, 2)
density(dist, 2, log = TRUE)

cdf(dist, 4)

quantile(dist, 0.7)

Create a distribution from p/d/q/r style functions

Description

If a distribution is not yet supported, you can vectorise p/d/q/r functions using this function. dist_wrap() stores the distributions parameters, and provides wrappers which call the appropriate p/d/q/r functions.

Using this function to wrap a distribution should only be done if the distribution is not yet available in this package. If you need a distribution which isn't in the package yet, consider making a request at https://github.com/mitchelloharawild/distributional/issues.

Usage

dist_wrap(dist, ..., package = NULL)

Arguments

Details

The dist_wrap() function provides a generic interface to create distribution objects from any set of p/d/q/r style functions. The statistical properties depend on the specific distribution being wrapped.

Examples

dist <- dist_wrap("norm", mean = 1:3, sd = c(3, 9, 2))

density(dist, 1) # dnorm()
cdf(dist, 4) # pnorm()
quantile(dist, 0.975) # qnorm()
generate(dist, 10) # rnorm()

library(actuar)
dist <- dist_wrap("invparalogis", package = "actuar", shape = 2, rate = 2)
density(dist, 1) # actuar::dinvparalogis()
cdf(dist, 4) # actuar::pinvparalogis()
quantile(dist, 0.975) # actuar::qinvparalogis()
generate(dist, 10) # actuar::rinvparalogis()

Extract the name of the distribution family

Description

Usage

## S3 method for class 'distribution'
family(object, ...)

Arguments

Examples

dist <- c(
  dist_normal(1:2),
  dist_poisson(3),
  dist_multinomial(size = c(4, 3),
  prob = list(c(0.3, 0.5, 0.2), c(0.1, 0.5, 0.4)))
  )
family(dist)

Randomly sample values from a distribution

Description

Generate random samples from probability distributions.

Usage

## S3 method for class 'distribution'
generate(x, times, ...)

Arguments

Check if a distribution is symmetric

Description

Determines whether a probability distribution is symmetric around its center.

Usage

has_symmetry(x, ...)

Arguments

Value

A logical value indicating whether the distribution is symmetric.

Examples

# Normal distribution is symmetric
has_symmetry(dist_normal(mu = 0, sigma = 1))
has_symmetry(dist_normal(mu = 5, sigma = 2))

# Beta distribution symmetry depends on parameters
has_symmetry(dist_beta(shape1 = 2, shape2 = 2))  # symmetric
has_symmetry(dist_beta(shape1 = 2, shape2 = 5))  # not symmetric

Compute highest density regions

Description

Used to extract a specified prediction interval at a particular confidence level from a distribution.

Usage

hdr(x, ...)

Arguments

Highest density regions of probability distributions

Description

This function is highly experimental and will change in the future. In particular, improved functionality for object classes and visualisation tools will be added in a future release.

Computes minimally sized probability intervals highest density regions.

Usage

## S3 method for class 'distribution'
hdr(x, size = 95, n = 512, ...)

Arguments

Compute intervals

Description

Used to extract a specified prediction interval at a particular confidence level from a distribution.

The numeric lower and upper bounds can be extracted from the interval using ⁠<hilo>$lower⁠ and ⁠<hilo>$upper⁠ as shown in the examples below.

Usage

hilo(x, ...)

Arguments

Examples

# 95% interval from a standard normal distribution
interval <- hilo(dist_normal(0, 1), 95)
interval

# Extract the individual quantities with `$lower`, `$upper`, and `$level`
interval$lower
interval$upper
interval$level

Probability intervals of a probability distribution

Description

Returns a hilo central probability interval with probability coverage of size. By default, the distribution's quantile() will be used to compute the lower and upper bound for a centered interval

Usage

## S3 method for class 'distribution'
hilo(x, size = 95, ...)

Arguments

See Also

hdr.distribution()

Test if the object is a distribution

Description

This function returns TRUE for distributions and FALSE for all other objects.

Usage

is_distribution(x)

Arguments

Value

TRUE if the object inherits from the distribution class.

Examples

dist <- dist_normal()
is_distribution(dist)
is_distribution("distributional")

Is the object a hdr

Description

Is the object a hdr

Usage

is_hdr(x)

Arguments

Is the object a hilo

Description

Is the object a hilo

Usage

is_hilo(x)

Arguments

Kurtosis of a probability distribution

Description

Usage

kurtosis(x, ...)

## S3 method for class 'distribution'
kurtosis(x, ...)

Arguments

The (log) likelihood of a sample matching a distribution

Description

Usage

likelihood(x, ...)

## S3 method for class 'distribution'
likelihood(x, sample, ..., log = FALSE)

log_likelihood(x, ...)

Arguments

Mean of a probability distribution

Description

Returns the empirical mean of the probability distribution. If the method does not exist, the mean of a random sample will be returned.

Usage

## S3 method for class 'distribution'
mean(x, ...)

Arguments

Median of a probability distribution

Description

Returns the median (50th percentile) of a probability distribution. This is equivalent to quantile(x, p=0.5).

Usage

## S3 method for class 'distribution'
median(x, na.rm = FALSE, ...)

Arguments

Construct distributions

Description

Allows extension package developers to define a new distribution class compatible with the distributional package.

Usage

new_dist(..., class = NULL, dimnames = NULL)

Arguments

Construct hdr intervals

Description

Construct hdr intervals

Usage

new_hdr(
  lower = list_of(.ptype = double()),
  upper = list_of(.ptype = double()),
  size = double()
)

Arguments

Value

A "hdr" vector

Author(s)

Mitchell O'Hara-Wild

Examples

new_hdr(lower = list(1, c(3,6)), upper = list(10, c(5, 8)), size = c(80, 95))

Construct hilo intervals

Description

Class constructor function to help with manually creating hilo interval objects.

Usage

new_hilo(lower = double(), upper = double(), size = double())

Arguments

Value

A "hilo" vector

Author(s)

Earo Wang & Mitchell O'Hara-Wild

Examples

new_hilo(lower = rnorm(10), upper = rnorm(10) + 5, size = 95)

Construct support regions

Description

Construct support regions

Usage

new_support_region(x = numeric(), limits = list(), closed = list())

Arguments

Extract the parameters of a distribution

Description

Usage

parameters(x, ...)

## S3 method for class 'distribution'
parameters(x, ...)

Arguments

Examples

dist <- c(
  dist_normal(1:2),
  dist_poisson(3),
  dist_multinomial(size = c(4, 3),
  prob = list(c(0.3, 0.5, 0.2), c(0.1, 0.5, 0.4)))
  )
parameters(dist)

Distribution Quantiles

Description

Computes the quantiles of a distribution.

Usage

## S3 method for class 'distribution'
quantile(x, p, ..., log = FALSE)

Arguments

Objects exported from other packages

Description

These objects are imported from other packages. Follow the links below to see their documentation.

generics

generate

Skewness of a probability distribution

Description

Usage

skewness(x, ...)

## S3 method for class 'distribution'
skewness(x, ...)

Arguments

Region of support of a distribution

Description

Usage

support(x, ...)

## S3 method for class 'distribution'
support(x, ...)

Arguments

Variance

Description

A generic function for computing the variance of an object.

Usage

variance(x, ...)

## S3 method for class 'numeric'
variance(x, ...)

## S3 method for class 'matrix'
variance(x, ...)

## S3 method for class 'numeric'
covariance(x, ...)

Arguments

Details

The implementation of variance() for numeric variables coerces the input to a vector then uses stats::var() to compute the variance. This means that, unlike stats::var(), if variance() is passed a matrix or a 2-dimensional array, it will still return the variance (stats::var() returns the covariance matrix in that case).

See Also

variance.distribution(), covariance()

Variance of a probability distribution

Description

Returns the empirical variance of the probability distribution. If the method does not exist, the variance of a random sample will be returned.

Usage

## S3 method for class 'distribution'
variance(x, ...)

Arguments