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\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.
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