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.

Title	Gumbel random variable
Canonical name	GumbelRandomVariable
Date of creation	2013-03-22 15:55:40
Last modified on	2013-03-22 15:55:40
Owner	georgiosl (7242)
Last modified by	georgiosl (7242)
Numerical id	4
Author	georgiosl (7242)
Entry type	Definition
Classification	msc 60E05