| Version 3 |
Version 2 |
| $X$ is a \emph{Weibull random variable} if it has a probability density function, given by |
$X$ is a \emph{Weibull random variable} if it has a probability density function, given by |
| $$f_X(x)=\frac{\gamma}{\alpha}(\frac{x-\mu}{\alpha})^{\gamma-1} |
$$f_X(x)=\frac{\gamma}{\alpha}(\frac{x-\mu}{\alpha})^{\gamma-1} |
| e^{-(\frac{x-\mu}{\alpha})^\gamma}$$ |
e^{-(\frac{x-\mu}{\alpha})^\gamma}$$ |
| 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}. |
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}. |
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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 distribution} |
| 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}: |
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| $$f_X(x)=\gamma x^{\gamma-1}\operatorname{exp}(-x^\gamma)$$ |
$$f_X(x)=\gamma x^{\gamma-1}\operatorname{exp}(-x^\gamma)$$ |
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| \textbf{\PMlinkescapetext{Properties}}: |
\textbf{\PMlinkescapetext{Properties}}: |
| Given a standard Weibull distribution $X\sim \mbox{Wei}(\gamma)$: |
Given a standard Weibull distribution $X\sim \mbox{Wei}(\gamma)$: |
| \begin{enumerate} |
\begin{enumerate} |
| \item |
\item |
| E[X]=$\Gamma(\frac{\gamma+1}{\gamma})$, where $\Gamma$ is the gamma function |
E[X]=$\Gamma(\frac{\gamma+1}{\gamma})$, where $\Gamma$ is the gamma function |
| \item |
\item |
| Median = $(\operatorname{ln}2)^{\frac{1}{\gamma}}$ |
Median = $(\operatorname{ln}2)^{\frac{1}{\gamma}}$ |
| \item |
\item |
| Mode $= \begin{cases} |
Mode $= \begin{cases} |
| (1-\frac{1}{\gamma})^{1/\gamma} & \mbox{if $\gamma>1$}\\ |
(1-\frac{1}{\gamma})^{1/\gamma} & \mbox{if $\gamma>1$}\\ |
| 0 & \mbox{otherwise} \end{cases}$ |
0 & \mbox{otherwise} \end{cases}$ |
| \item |
\item |
| Var[X]=$\sqrt{\Gamma(\frac{\gamma+2}{\gamma})-\Gamma(\frac{\gamma+1}{\gamma})^2}$ |
Var[X]=$\sqrt{\Gamma(\frac{\gamma+2}{\gamma})-\Gamma(\frac{\gamma+1}{\gamma})^2}$ |
| \item |
\item |
| $X\sim \mbox{Wei}(\alpha,\gamma,0)$ iff $X^{\gamma}\sim \mbox{Exp}(\alpha^\gamma)$, the exponential distribution with parameter $\alpha^\gamma$ |
$X\sim \mbox{Wei}(\alpha,\gamma,0)$ iff $X^{\gamma}\sim \mbox{Exp}(\alpha^\gamma)$, the exponential distribution with parameter $\alpha^\gamma$ |
| \end{enumerate} |
\end{enumerate} |
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| \textbf{Remark.} |
\textbf{Remark.} |
| The Weibull distribution is often used to model reliability or lifetime of \PMlinkescapetext{products} such as light bulbs. |
The Weibull distribution is often used to model reliability or lifetime of \PMlinkescapetext{products} such as light bulbs. |
