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 $\alpha ,\gamma ,\mu \in ℝ$, $\alpha ,\gamma >0$ and $x\ge \mu$. $\alpha$ is the scale parameter, $\gamma$ is the shape parameter, and $\mu$ is the location parameter.

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

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

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

$$f_X(x)=\gamma x^{\gamma -1}\exp (-x^{\gamma })$$

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

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

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

3. Mode

4. $\mathrm{Var}[X]=\mathrm{\Gamma }(\frac{\gamma +2}{\gamma })-\mathrm{\Gamma }{(\frac{\gamma +1}{\gamma })}^2$

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

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