Create a Gumbel distribution

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.

Value

A Gumbel object.

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.

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
#> [1] "Gumbel distribution (mu = 1, sigma = 2)"

random(X, 10)
#>  [1]  8.104751940 -0.816379582  5.007573903  0.789488808  0.183959497
#>  [6]  1.183838833 -0.929543900 -2.587372533 -0.373340977 -0.002439646

pdf(X, 0.7)
#> [1] 0.1817758
log_pdf(X, 0.7)
#> [1] -1.704981

cdf(X, 0.7)
#> [1] 0.3129117
quantile(X, 0.7)
#> [1] 3.061861

cdf(X, quantile(X, 0.7))
#> [1] 0.7
quantile(X, cdf(X, 0.7))
#> [1] 0.7