# Weibull random variable

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

 $f_{X}(x)=\frac{\gamma}{\alpha}(\frac{x-\mu}{\alpha})^{\gamma-1}e^{-(\frac{x-% \mu}{\alpha})^{\gamma}}$

where $\alpha,\gamma,\mu\in\mathbb{R}$, $\alpha,\gamma>0$ and $x\geq\mu$. $\alpha$ is the scale parameter, $\gamma$ is the shape parameter, and $\mu$ is the location parameter.

Notation for $X$ having a Weibull distribution is $X\sim\mbox{Wei}(\alpha,\gamma,\mu)$. Usually, the location and scale parameters are dropped by the transformation

 $Y=\frac{X-\mu}{\alpha}$

so that $Y\sim\mbox{Wei}(\gamma):=\mbox{Wei}(1,\gamma,0)$. The resulting distribution is called the standard Weibull, or Rayleigh distribution:

 $f_{X}(x)=\gamma x^{\gamma-1}\operatorname{exp}(-x^{\gamma})$

: Given a standard Weibull distribution $X\sim\mbox{Wei}(\gamma)$:

1. 1.

$\operatorname{E}[X]=\Gamma(\frac{\gamma+1}{\gamma})$, where $\Gamma$ is the gamma function

2. 2.

Median = $(\operatorname{ln}2)^{\frac{1}{\gamma}}$

3. 3.

Mode $=\begin{cases}(1-\frac{1}{\gamma})^{1/\gamma}&\mbox{if \gamma>1}\\ 0&\mbox{otherwise}\end{cases}$

4. 4.

$\operatorname{Var}[X]=\Gamma(\frac{\gamma+2}{\gamma})-\Gamma(\frac{\gamma+1}{% \gamma})^{2}$

5. 5.

$X\sim\mbox{Wei}(\alpha,\gamma,0)$ iff $X^{\gamma}\sim\mbox{Exp}(\alpha^{\gamma})$, the exponential distribution with parameter $\alpha^{\gamma}$

Remark. The Weibull distribution is often used to model reliability or lifetime of such as light bulbs.

Title Weibull random variable WeibullRandomVariable 2013-03-22 14:26:44 2013-03-22 14:26:44 CWoo (3771) CWoo (3771) 8 CWoo (3771) Definition msc 62N99 msc 62E15 msc 60E05 msc 62P05 Weibull distribution Rayleigh distribution