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