Weibull random variable
WeibullRandomVariable
2004-06-24 19:55:04
2007-06-06 11:40:44
Definition
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$X$ is a \emph{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 $x\ge\mu$. $\alpha$ is the \emph{scale parameter}, $\gamma$ is the \emph{shape parameter}, and $\mu$ is the \emph{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 \emph{standard Weibull}, or \emph{Rayleigh distribution}:
$$f_X(x)=\gamma x^{\gamma-1}\operatorname{exp}(-x^\gamma)$$
\textbf{\PMlinkescapetext{Properties}}:
Given a standard Weibull distribution $X\sim \mbox{Wei}(\gamma)$:
\begin{enumerate}
\item
$\operatorname{E}[X]=\Gamma(\frac{\gamma+1}{\gamma})$, where $\Gamma$ is the gamma function
\item
Median = $(\operatorname{ln}2)^{\frac{1}{\gamma}}$
\item
Mode $= \begin{cases}
(1-\frac{1}{\gamma})^{1/\gamma} & \mbox{if $\gamma>1$}\\
0 & \mbox{otherwise} \end{cases}$
\item
$\operatorname{Var}[X]=\Gamma(\frac{\gamma+2}{\gamma})-\Gamma(\frac{\gamma+1}{\gamma})^2$
\item
$X\sim \mbox{Wei}(\alpha,\gamma,0)$ iff $X^{\gamma}\sim \mbox{Exp}(\alpha^\gamma)$, the exponential distribution with parameter $\alpha^\gamma$
\end{enumerate}
\textbf{Remark.}
The Weibull distribution is often used to model reliability or lifetime of \PMlinkescapetext{products} such as light bulbs.