

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

\begin{slide}
\begin{itemize}
\item Then $(\alpha )$ from additivity of binomial distributions; $y$
is binomial with distribution $b(k,n,p)$,

\item $(\beta)$ from standard central limit theorem, $y$ is Gaussian
with mean $n\mu$ and standard deviation $\sqrt{n}\sigma$.

\item Here $\mu$ and
$\sigma$ are mean and standard deviation of $b(k,1,p)$ distribution,

\item i.e.,
\begin{eqnarray*}
\mu & = & 0 \cdot (1-p) + 1\cdot p = p \\
\sigma^2 & = & \left\langle x^2 \right\rangle - \langle x\rangle^2 \\
& = & 0 \cdot (1-p) + 1\cdot p - p^2 = p(1-p)
\end{eqnarray*}

\item We conclude that for fixed $p$, and $n\rightarrow\infty$, $b(k,n,p)$
tends to Gaussian distribution with mean $np$ and standard deviation
$\sqrt{p(1-p)n}$.
\end{itemize}
\end{slide}

