Type-2 Gumbel distribution

Type-2 Gumbel

Parameters	${\displaystyle a\!}$ (real) ${\displaystyle b\!}$ shape (real)
PDF	${\displaystyle abx^{-a-1}e^{-bx^{-a}}\!}$
CDF	${\displaystyle e^{-bx^{-a}}\!}$
Mean	${\displaystyle b^{1/a}\Gamma (1-1/a)\!}$
Variance	${\displaystyle b^{2/a}(\Gamma (1-1/a)-{\Gamma (1-1/a)}^{2})\!}$

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

${\displaystyle f(x|a,b)=abx^{-a-1}e^{-bx^{-a}}\,}$

for

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

For ${\displaystyle 0<a\leq 1}$ the mean is infinite. For ${\displaystyle 0<a\leq 2}$ the variance is infinite.

The cumulative distribution function is

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

The moments ${\displaystyle E[X^{k}]\,}$ exist for ${\displaystyle k<a\,}$

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

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

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

Related distributions

• The special case b = 1 yields the Fréchet distribution.
• Substituting ${\displaystyle b=\lambda ^{-k}}$ and ${\displaystyle a=-k}$ 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.

Based on The GNU Scientific Library, used under GFDL.

See also

• Extreme value theory
• Gumbel distribution