Gumbel random variable
$X$ is a Gumbel random variable if it has a probability density function^{}, given by
$${f}_{X}(x)=\frac{1}{\sigma}\mathrm{exp}(\frac{x\mu}{\sigma})S(x)$$ 
where $$, $\mu $ is the location parameter, $\sigma $ is the scale parameter, and $S(x)$ is the survivor function, $S(x)=\mathrm{exp}[\mathrm{exp}(\frac{x\mu}{\sigma})]$ .
Notation for $X$ having a Gumbel distribution is $X\sim \text{Gum}(\mu ,\sigma )$.
: Given a Gumbel distribution $X\sim \text{Gum}(\mu ,\sigma )$:

1.
E[X]=$\mu \gamma \sigma $, where $\gamma $ is the Euler’s constant

2.
Var[X]=$\frac{{\pi}^{2}}{6}{\sigma}^{2}$
Remark. Nevertheless the interval $(\mathrm{\infty},\mathrm{\infty})$ in which is defined, the Gumbel distribution is often used to model reliability or lifetime of products.
Title  Gumbel random variable 

Canonical name  GumbelRandomVariable 
Date of creation  20130322 15:55:40 
Last modified on  20130322 15:55:40 
Owner  georgiosl (7242) 
Last modified by  georgiosl (7242) 
Numerical id  4 
Author  georgiosl (7242) 
Entry type  Definition 
Classification  msc 60E05 