1 2\begin{tabular}{lcll}\hline 3Variate & $x$ & \ccode{double} & $\mu \leq x < \infty$ \\ 4Location & $\mu$ & \ccode{double} & $-\infty < \mu < \infty$\\ 5Scale & $\lambda$ & \ccode{double} & $\lambda > 0$ \\ \hline 6\end{tabular} 7 8The probability density function (PDF) is: 9 10\begin{equation} 11P(X=x) = \lambda e^{-\lambda (x - \mu)} 12\end{equation} 13 14The cumulative distribution function (CDF) is: 15 16\begin{equation} 17P(X \leq x) = 1 - e^{-\lambda (x - \mu)} 18\end{equation} 19 20 21\subsection{Sampling} 22 23An exponentially distributed sample $x$ is generated by the 24transformation method, using the fact that if $R$ is uniformly 25distributed on $(0,1]$, $1-R$ is uniformly distributed on $[0,1)$: 26 27\[ 28 R = \mbox{uniform positive sample in (0,1]}\\ 29 x = \mu - \frac{1}{lambda} \log(R) 30\] 31 32\subsection{Maximum likelihood fitting} 33 34The maximum likelihood estimate $\hat{\lambda}$ is $\frac{1}{\sum_i 35x_i}$. The distribution of $\frac{\lambda}{\hat{\lambda}}$ is 36approximately normal with mean 1 and standard error $\frac{1}{\sqrt{N}}$ 37\citep{Lawless82}. 38 39% xref J1/p49 for derivation of standard error. 40 41 42