Type-2 Gumbel distribution

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


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

for


 * $$0 < x < \infty$$.

This implies that it 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 $$0<a\le 1$$ the mean is infinite. For $$0<a\le 2$$ the variance is infinite.

The cumulative distribution function is


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

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

The special case b = 1 yelds the Fréchet distribution

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