Gumbel Distribution

Definition

$X\sim \mathrm{Gumbel}(\mu ,\beta )$ where $\mu \in \mathbb{R}$ is Location Parameter and $\beta \in {\mathbb{R}}^{+}$ is Scale Parameter.

Standard Gumbel Distribution

$X\sim \mathrm{Gumbel}(0,1)$ $f(x)=\exp [-(x+\exp (-x))]$

Properties

PDF

$f(x)=\frac{1}{\beta }\exp \left[-\left(\frac{x-\mu }{\beta }+\exp (\frac{x-\mu }{\beta })\right)\right],x\in \mathbb{R}$

Mean

$E(X)=\mu$

Variance

$\mathrm{Var}(X)=\frac{{\pi }^2}{6}{\beta }^2$