\begin{tabular}{lcll}\hline Variate & $x$ & \ccode{double} & $\mu \leq x < \infty$ \\ Location & $\mu$ & \ccode{double} & $-\infty < \mu < \infty$\\ Scale & $\lambda$ & \ccode{double} & $\lambda > 0$ \\ \hline \end{tabular} The probability density function (PDF) is: \begin{equation} P(X=x) = \lambda e^{-\lambda (x - \mu)} \end{equation} The cumulative distribution function (CDF) is: \begin{equation} P(X \leq x) = 1 - e^{-\lambda (x - \mu)} \end{equation} \subsection{Sampling} An exponentially distributed sample $x$ is generated by the transformation method, using the fact that if $R$ is uniformly distributed on $(0,1]$, $1-R$ is uniformly distributed on $[0,1)$: $R = \mbox{uniform positive sample in (0,1]}\\ x = \mu - \frac{1}{lambda} \log(R)$ \subsection{Maximum likelihood fitting} The maximum likelihood estimate $\hat{\lambda}$ is $\frac{1}{\sum_i x_i}$. The distribution of $\frac{\lambda}{\hat{\lambda}}$ is approximately normal with mean 1 and standard error $\frac{1}{\sqrt{N}}$ \citep{Lawless82}. % xref J1/p49 for derivation of standard error.