Gumbel: Create a Gumbel distribution

In distributions3: Probability Distributions as S3 Objects

Create a Gumbel distribution

Description

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

Usage

Gumbel(mu = 0, sigma = 1)

Arguments

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

Details

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

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

Support: R, the set of all real numbers.

Mean: μ + σγ, where γ is Euler's constant, approximately equal to 0.57722.

Median: μ - σ ln(ln 2).

Variance: σ^2 π^2 / 6.

Probability density function (p.d.f):

f(x) = (1 / σ) exp[-(x - μ) / σ] exp{-exp[-(x - μ) / σ]}

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

Cumulative distribution function (c.d.f):

In the ξ = 0 (Gumbel) special case

F(x) = exp{ - exp[-(x - μ) / σ]}

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

Value

A Gumbel object.

See Also

Other continuous distributions: Beta(), Cauchy(), ChiSquare(), Erlang(), Exponential(), FisherF(), Frechet(), GEV(), GP(), Gamma(), LogNormal(), Logistic(), Normal(), RevWeibull(), StudentsT(), Tukey(), Uniform(), Weibull()

Examples

set.seed(27)

X <- Gumbel(1, 2)
X

random(X, 10)

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

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

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

distributions3 documentation built on Sept. 7, 2022, 5:07 p.m.