Gumbel {distributions3}R Documentation

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.


Gumbel(mu = 0, sigma = 1)



The location parameter, written \mu in textbooks. mu can be any real number. Defaults to 0.


The scale parameter, written \sigma in textbooks. sigma can be any positive number. Defaults to 1.


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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.


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()



X <- Gumbel(1, 2)

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))

[Package distributions3 version 0.2.1 Index]