distributional (version 0.2.1)

dist_gumbel: The Gumbel distribution

Description

stable

Usage

dist_gumbel(alpha, scale)

Arguments

alpha

location parameter.

scale

parameter. Must be strictly positive.

Details

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.

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

# NOT RUN {
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)

generate(dist, 10)

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

cdf(dist, 4)

quantile(dist, 0.7)

# }