# Type-2 Gumbel distribution

Parameters $a\!$ (real) $b\!$ shape (real) $a b x^{-a-1} e^{-b x^{-a}}\!$ $e^{-b x^{-a}}\!$

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

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

for

$0 < x < \infty$.

This implies that it is similar to the Weibull distributions, substituting $b=\lambda^{-k}$ and $a=-k$. 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 $0 the mean is infinite. For $0 the variance is infinite.

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

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

The special case b = 1 yields the Fréchet distribution

Based on The GNU Scientific Library, used under GFDL.