

\subsection{\bf Gaussian Limit of Binomial}
\subsection{\bf and Poisson Distributions---IV}

\begin{slide}
\begin{itemize}
\item Standard deviation of Poisson is given by
\begin{eqnarray*}
\sigma^2 & = & \left\langle x^2 \right\rangle - \lambda^2 \\
& = & e^{-\lambda} \left( \frac{\lambda\partial}{\partial\lambda}
\right)^2 \left( e^\lambda \right) - \lambda^2 \\
& = & e^{-\lambda} \left( \frac{\lambda\partial}{\partial\lambda}
\right) \left(\lambda e^\lambda \right) - \lambda^2 \\
& = & \lambda + \lambda^2 - \lambda^2 = \lambda = \mu
\end{eqnarray*}

\item So, for very large $\Lambda$, any Poissonly distributed random
variable $y$ has mean $\Lambda$ and standard deviation
$\sqrt{\Lambda}$.

\item We will soon show that counts in any counting experiment (e.g.,
number of raindrops in a bowl or particles in a high energy physical
experiment) is described by a Poisson random variable with (large)
parameter $\Lambda = \mbox{number}$ of expected counts.

\item The above result then translates into well known result that
counting experiment with $N$ events has error $\sqrt{N}$.
\end{itemize}
\end{slide}

