Type-2 Gumbel distribution

Type-2 Gumbel

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

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

The cumulative distribution function is

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

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

The special case b = 1 yields the Fréchet distribution

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Based on The GNU Scientific Library, used under GFDL.

See also

• Extreme value theory
• Gumbel distribution
• Type-1 Gumbel distribution