# Gumbel random variable

$X$ is a Gumbel random variable if it has a probability density function, given by

 $f_{X}(x)=\frac{1}{\sigma}\exp(\frac{x-\mu}{\sigma})S(x)$

where $-\infty, $\mu$ is the location parameter, $\sigma$ is the scale parameter, and $S(x)$ is the survivor function, $S(x)=\exp[-\exp(\frac{x-\mu}{\sigma})]$ .

Notation for $X$ having a Gumbel distribution is $X\sim\mbox{Gum}(\mu,\sigma)$.

: Given a Gumbel distribution $X\sim\mbox{Gum}(\mu,\sigma)$:

1. 1.

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

2. 2.

Var[X]=$\frac{\pi^{2}}{6}\sigma^{2}$

Remark. Nevertheless the interval $(-\infty,\infty)$ in which is defined, the Gumbel distribution is often used to model reliability or lifetime of products.

Title Gumbel random variable GumbelRandomVariable 2013-03-22 15:55:40 2013-03-22 15:55:40 georgiosl (7242) georgiosl (7242) 4 georgiosl (7242) Definition msc 60E05