dist_gumbel {distributional}

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

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 = \mu, scale parameter sigma = \sigma.

Support: R, the set of all real numbers.

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

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

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

Probability density function (p.d.f):

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

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

Cumulative distribution function (c.d.f):

In the \xi = 0 (Gumbel) special case

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

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

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)