Type-2 Gumbel distribution

Short description: Probability distribution

Type-2 Gumbel

Parameters	$a$ (real) $b$ shape (real)
PDF	$ab{x}^{-a-1}{e}^{-b{x}^{-a}}$
CDF	$e^{-b{x}^{-a}}$
Mean	$b^{1/a}\mathrm{\Gamma }(1-1/a)$
Variance	$b^{2/a}(\mathrm{\Gamma }(1-1/a)-{\mathrm{\Gamma }(1-1/a)}^2)$

In probability theory, the Type-2 Gumbel probability density function is

$f(x|a,b)=ab{x}^{-a-1}{e}^{-b{x}^{-a}}$

for

$0<x<\mathrm{\infty }$.

For $0<a\le 1$ the mean is infinite. For $0<a\le 2$ the variance is infinite.

The cumulative distribution function is

$F(x|a,b)=e^{-b{x}^{-a}}$

The moments $E[X^k]$ exist for $k<a$

The distribution is named after Emil Julius Gumbel (1891 – 1966).

Generating random variates

Given a random variate U drawn from the uniform distribution in the interval (0, 1), then the variate

$X=(-\ln U/b{)}^{-1/a},$

has a Type-2 Gumbel distribution with parameter $a$ and $b$. This is obtained by applying the inverse transform sampling-method.

Related distributions

• The special case b = 1 yields the Fréchet distribution.
• Substituting $b={\lambda }^{-k}$ and $a=-k$ yields the Weibull distribution. Note, however, that a positive k (as in the Weibull distribution) would yield a negative a and hence a negative probability density, which is not allowed.

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Based on The GNU Scientific Library, used under GFDL.

See also

• Extreme value theory
• Gumbel distribution