

\begin{slide}
\subsection{\bf Poisson Limit of Binomial Distribution}
{\Smallsize
\begin{itemize}
\item The Poisson Limit is a very important special case of the binomial
distribution, and it describes anything from telephone calls to events
observed in particle physics.

\item Take limit in $b(k,n,p)$ of $n\rightarrow\infty$ and
$p\rightarrow 0$ such that $np=\lambda$ remains finite.

\item $k$ is also fixed.
\[
b\left( k, n, \frac{\lambda}{n} \right) = \frac{n!}{k! (n-k)!} \left(
\frac{\lambda}{n} \right)^k \left( 1 - \frac{\lambda}{n} \right)^n
\]

\item Now $\frac{n!}{(n-k)!} = n(n-1) \ldots (n-k+1) \rightarrow n^k$ as
$n\rightarrow \infty$ for fixed $k$.  (You can also get this using
Stirling's formula for the gamma function).  

\item Also,
\begin{eqnarray*}
\left( 1 - \frac{\lambda}{n} \right)^{n-k} & \approx & \left( 1 -
\frac{\lambda}{n}\right)^n \rightarrow e^{-\lambda}. \hspace{1em} \mbox{So,}
\nonumber \\
b \left( k, n, \frac{\lambda}{n} \right) & \rightarrow & \frac{n^k}{k!}\,
\frac{\lambda^k}{n^k}\, e^{-\lambda} \nonumber \\
& = & P(k,\lambda) = e^{-\lambda} \frac{\lambda^k}{k!}
\end{eqnarray*}

\item which is the Poisson Distribution $P(k,\lambda)$ of $k$ events.

\item Note: ${\displaystyle \sum^\infty_{k=0}} P(k,\lambda)=1$ as it should.
\end{itemize}
}
\end{slide}

