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 The Weibull distribution may be useful for fitting fat-tailed empirical distributions. In the literature, the Weibull is sometimes called a stretched exponential'' distribution when its shape parameter $\tau$ is less than 1. Stretched exponential'' distributions in the literature are either Weibull (PDF $= \lambda \tau (\lambda x)^\tau exp\left[- (\lambda x)^tau \right]$ or a more simple PDF $\propto exp\left[- {\lambda(x-\mu)}^tau \right]$. Easel treats the latter form in the \eslmod{stretchexp} module. \subsection{Weibull densities} The probability density function (PDF) is: \begin{equation} P(X=x) = \lambda \tau [\lambda(x - \mu)]^{\tau-1} e^{- [\lambda(x-\mu)]^{\tau}} \label{eqn:weibull_pdf} \end{equation} The cumulative distribution function (CDF) is: \begin{equation} P(X \leq x) = 1 - e^{- [\lambda(x-\mu)]^{\tau}} \label{eqn:weibull_cdf} \end{equation} Variate $x$ ranges $\mu \leq x < \infty$. (However, for $\tau < 1$, the PDF goes to infinity at $x=\mu$, so evaluating at $x=\mu$ may not be desired.) Location parameter $\mu$ is unconstrained, $-\infty < \mu < \infty$. (Weibull distributions are usually represented without an explicit location parameter, implicitly assuming $\mu = 0$.) Scale parameter $\lambda$ is nonnegative, $\lambda > 0$. (Alteratively, Weibull distributions are also sometimes represented with a scale parameter $b = \frac{1}{\lambda}$.) Shape parameter $\tau$ is nonnegative, $\tau > 0$.