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

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

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

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

Given a random variate ${\displaystyle \ U\ }$ drawn from the uniform distribution in the interval ${\displaystyle \ (0,1)\ ,}$ then the variate

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

Based on "Gumbel distribution". The GNU Scientific Library. type 002d2, used under GFDL.