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 \xi 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 \mu in textbooks. mu can be any real number. Defaults to 0.
sigma	The scale parameter, written \sigma 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 = \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.

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. 30, 2024, 9:37 a.m.