Type-2 Gumbel distribution

\ f(x|a,b)=a\ b\ x^{-a-1}\ e^{-b\ x^{-a}}\quad

for

\quad x>0~.

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

\ F(x|a,b)=e^{-b\ x^{-a}}~.

The moments $\ \mathbb {E} {\bigl [}X^{k}{\bigr ]}\$ 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=\left(-{\frac {\ln U}{b}}\right)^{-{\frac {1}{a}}}\

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

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.

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

Parameters	$\ a\in \mathbb {R} \$ (shape), $\ b\in \mathbb {R} \$ (scale)
Support	$\ 0<x<\infty \$
PDF	$\ a\ b\ x^{-a-1}\ e^{-b\ x^{-a}}\$
CDF	$\ e^{-b\ x^{-a}}\$
Quantile	$\ \left(-\ {\frac {\ \log _{e}\!\left(p\right)\ }{b}}\right)^{-{\frac {1}{a}}}\$
Mean	$\ b^{\frac {1}{a}}\ \Gamma \!\left(\ 1-{\tfrac {\ 1\ }{a}}\ \right)\$
Variance	$\ b^{\frac {2}{a}}\ \Gamma \!\left(1-{\tfrac {\ 1\ }{a}}\ \right){\Bigl (}1-\Gamma \!\left(1-{\tfrac {1}{a}}\right){\Bigr )}\$