

\subsection{\bf Example of Poisson Distribution}

\begin{slide}
\begin{itemize}
\item Given circumstances of previous Binomial example, what is
chance that Frank Nagy smiles exactly once in a {\em given\/} lecture:

\item {\bf Solution:}\ Again, we set each trial = a smile and we have 1,000
of them.  Now Success = ``Frank Nagy smiled this smile in our given
lecture'' and it is not very probable $p=\frac{1}{20.50} =
\frac{1}{1000}$.  So, $n$ is large, $p$ is small, but $np=\lambda=1$ is
reasonable.

\item So, the Poisson approximation to the exact (in this example) Binomial
distribution is valid and chance is $e^{-1}$, i.e., it is quite
probable that Frank smiles once and once only in a given lecture.

\item The binomial estimate is
\[
\left( \begin{array}{c} 1000 \\ 1 \end{array} \right) \frac{1}{1000}
\left( \frac{999}{1000} \right)^{999} =
\left\lbrack \frac{999}{1000} \right\rbrack^{999}
\]
which is indeed $\approx e^{-1}$.
\end{itemize}
\end{slide}

