# Type-2 Gumbel distribution

Parameters Probability density function Cumulative distribution function ${\displaystyle a\!}$ (real)${\displaystyle b\!}$ shape (real) ${\displaystyle abx^{-a-1}\exp(-bx^{-a})\!}$ ${\displaystyle \exp(-bx^{-a})\!}$

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

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

for

${\displaystyle 0.

This implies that it 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.

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

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

The special case b = 1 yelds the Fréchet distribution

Based on gsl-ref_19.html#SEC309, used under GFDL.