

\subsection{\bf Moment Generating Function of a}
\subsection{\bf Sum of Two Random Variables}

\begin{slide}

\begin{itemize}
\item In an important application of moment generating functions, put
$h=x+y$ where $x$ and $y$ are independent.

\item From joint probability distribution, it follows that the distribution
density of $h$ is
\[
w(h) = \int\limits^{+\infty}_{-\infty} \int\limits^{+\infty}_{-\infty} 
\delta (h-x-y)\, p(x)\, q(y)\ dx\, dy
\]
\begin{eqnarray*} 
\varphi_h (k) & = & \int\limits^{+\infty}_{-\infty} dh\, w(h) e^{ikh} \\
& = & \int\limits^{+\infty}_{-\infty} \int\limits^{+\infty}_{-\infty} 
e^{ik(x+y)}\, p(x)\, q(y)\ dx\, dy \\
& = & \varphi_x (k) \varphi_y (k)
\end{eqnarray*}

\item So characteristic functions multiply for addition of two
independent random variables.

\item This result can obviously be extended to sum of $n$ independent
random variables.
\end{itemize}
\end{slide}

