Package 'distributional'

Description

Usage

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

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

Arguments

Description

A generic function for computing the covariance of an object.

Usage

covariance(x, ...)

Arguments

See Also

covariance.distribution(), variance()

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

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

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)

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)

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)

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)

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)

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)

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)

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)

Description

The Dirichlet distribution is a multivariate generalisation of the Beta distribution. It is the conjugate prior of the Categorical and Multinomial distributions, and describes a probability distribution over the $(k-1)$-simplex — the set of $k$-dimensional vectors whose components are non-negative and sum to one.

Usage

dist_dirichlet(alpha)

Arguments

Details

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

In the following, let $\mathbf{X} = (X_1, \ldots, X_k)$ be a Dirichlet random variable with concentration parameter alpha = $\boldsymbol{\alpha} = (\alpha_1, \ldots, \alpha_k)$, where each $\alpha_i > 0$.

Support: $\mathbf{x}$ on the $(k-1)$-simplex, i.e. $x_i \geq 0$ and $\sum_{i=1}^k x_i = 1$.

Mean: $E(X_i) = \frac{\alpha_i}{\alpha_0}$ where $\alpha_0 = \sum_{i=1}^k \alpha_i$.

Variance:

$\mathrm{Var}(X_i) = \frac{\alpha_i(\alpha_0 - \alpha_i)}{\alpha_0^2(\alpha_0 + 1)}$

Covariance:

$\mathrm{Cov}(X_i, X_j) = \frac{-\alpha_i \alpha_j}{\alpha_0^2(\alpha_0 + 1)}, \quad i \neq j$

Probability density function (p.d.f):

$f(\mathbf{x}) = \frac{1}{B(\boldsymbol{\alpha})} \prod_{i=1}^k x_i^{\alpha_i - 1}$

where $B(\boldsymbol{\alpha}) = \frac{\prod_{i=1}^k \Gamma(\alpha_i)}{\Gamma(\alpha_0)}$ is the multivariate Beta function.

See Also

LaplacesDemon::ddirichlet(), LaplacesDemon::rdirichlet()

Examples

dist <- dist_dirichlet(alpha = list(c(2, 5, 3)))
dist


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

density(dist, cbind(0.2, 0.5, 0.3))
density(dist, cbind(0.2, 0.5, 0.3), log = TRUE)

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)

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)

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)

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)

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)

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)

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)

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)

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.