

\subsection{\bf Additivity Property of Binomial}
\subsection{\bf or Poisson Distributions}

\begin{slide}
\begin{itemize}
\item These two distributions (Binomial and Poisson) share an important
additivity property, which is

\item $\alpha)$, obvious and $\beta)$, leads to confusion when used in
conjunction with central limit theorem (see later).

\item (a) Suppose the independent random variables $x_1$ and $x_2$
have binomial distributions with parameters $(n_1, p)$ and $(n_2, p)$
respectively.  

\begin{description}
\item{$\bullet$} Here, $x_i = k$ if there are $k$ successes.  
\item{$\bullet$} Then it is obvious from source of binomial
distribution that $x_1 + x_2$ has a binomial distribution with
parameter $(n_1 + n_2, p)$.
\end{description}

\item (b) Suppose independent random variables $y_1$ and $y_2$ have
Poisson distributions $\lambda_1$ and $\lambda_2$.  

\begin{description}
\item{$\bullet$} Then $y_1 + y_2$ has Poisson distribution with
parameter $\lambda_1 + \lambda_2$.
\item{$\bullet$} This is obvious either algebraically, or by using 
\item{$\bullet$} the result (a) above with $n_1=\lambda_2/p$ and
$\lambda_2 = \lambda_2/p$ and let $n_i\rightarrow\infty$, $p\rightarrow
0$, fixed $\lambda_1$, $\lambda_2$.
\end{description}
\end{itemize}
\end{slide}

