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 , $\mu$ 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 $X\sim \text{Gum}(\mu ,\sigma )$.

: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.