

\subsection{\bf Compound Poisson Distribution}
\begin{slide}
\begin{itemize}
\item Let $y={\displaystyle \sum^N_{i=1}} y_i$

\item $y_i$ is Poisson with mean $\mu$

\item $N$ is Poisson with mean $\lambda$

\item $Pr (y = \mbox{integer}\ r)$ is ${\displaystyle
\sum^\infty_{N=0}} Pr (N \mbox{takes given value}) \times Pr \left(
{\displaystyle \sum^N_{i=1}} y_i (N\ \mbox{fixed has $r$ events})
\right)$.

\item As for fixed $N$, $y$ is Poisson with mean $N\mu$, we find
\begin{eqnarray*}
Pr & ( & y = \mbox{integer}\ r ) \\
& = & {\displaystyle \sum^\infty_{N=0}} \frac{\lambda^N 
e^{-\lambda}}{\lambda!} \cdot \frac{(N\mu)^r e^{-N\mu}}{r!}
\end{eqnarray*}

\item More generally, we could give $y_i$ a non-Poisson distribution.
One example of non-Poisson $y_i$, is: 

\item The physics of charged particle ionization---p.53 EDJRS
\end{itemize}
\end{slide}

