# Type-2 Gumbel distribution

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

Parameters ${\displaystyle a\!}$ (real)${\displaystyle b\!}$ shape (real) ${\displaystyle abx^{-a-1}e^{-bx^{-a}}\!}$ ${\displaystyle e^{-bx^{-a}}\!}$ ${\displaystyle b^{1/a}\Gamma (1-1/a)\!}$ ${\displaystyle b^{2/a}(\Gamma (1-1/a)-{\Gamma (1-1/a)}^{2})\!}$
${\displaystyle f(x|a,b)=abx^{-a-1}e^{-bx^{-a}}\,}$

for

${\displaystyle 0.

This implies that it is similar to the Weibull distributions, substituting ${\displaystyle b=\lambda ^{-k}}$ and ${\displaystyle a=-k}$. Note, however, that a positive k (as in the Weibull distribution) would yield a negative a, which is not allowed here as it would yield a negative probability density.

For ${\displaystyle 0 the mean is infinite. For ${\displaystyle 0 the variance is infinite.

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

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

The special case b = 1 yields the Fréchet distribution.

Based on The GNU Scientific Library, used under GFDL.