dist_gumbel {distributional}R Documentation

The Gumbel distribution

Description

[Stable]

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

dist_gumbel(alpha, scale)

Arguments

alpha

location parameter.

scale

parameter. Must be strictly positive.

Details

We recommend reading this documentation on https://pkg.mitchelloharawild.com/distributional/, where the math will render nicely.

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.

See Also

actuar::Gumbel

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)


[Package distributional version 0.3.2 Index]