Type-2 Gumbel distribution

Probability distribution
name =Type-2 Gumbel| type =density
pdf_

cdf_

parameters =$a!$ (real)
$b!$ shape (real)
support =
pdf =$a\; b\; x^\{-a-1\}\; e^\{-b\; x^\{-a$!
cdf =$e^\{-b\; x^\{-a!$
mean =
median =
mode =
variance =
skewness =
kurtosis =
entropy =
mgf =
char =
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

:.

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

The cumulative distribution function is

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

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

The special case b = 1 yields the Fréchet distribution

----Based on [http://www.gnu.org/software/gsl/manual/gsl-ref_19.html#SEC309 gsl-ref_19.html#SEC309] , used under GFDL.

ee also

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