$X$ 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 <x<\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 $X\sim \mbox{Gum}(\mu,\sigma)$ .

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

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