

\subsection{\bf Moment Generating Function III}

\begin{slide}
\begin{itemize}
\item Expanding the exponential in definition of $\varphi (k)$ give
\[
\varphi (k) = \sum_n k^n E \frac{(ix)^n}{n!}
\]

\item where $E$ is the expectation or mean value.
\[
\varphi (k) = 1 + i\langle x\rangle k - \frac{\langle x^2 \rangle k^2}{2}
, \mbox{etc.}
\]

\item For a Gaussian distribution
\begin{eqnarray*}
\varphi (k) & = & \frac{1}{\sqrt{2\pi}\sigma} \int\limits^{+\infty}_{-\infty} 
\exp (ixk) \exp \left\lbrack - \frac{1}{2} 
\frac{(x-\mu)^2}{\sigma^2}\right\rbrack \\
& = & \exp \left\lbrace i\mu k - \frac{1}{2} k^2 \sigma^2 \right\rbrace
\end{eqnarray*}

\item EJDRS define the Cumulant as
\[
K(k) = \ln \varphi (k)
\]

\item I have never used this!  Note that
\[
K(k) = \langle x\rangle ik - \underbrace{\left( \left\langle x^2 \right\rangle
- \langle x\rangle^2 \right)}_{\sigma^2} \frac{k^2}{2}
\]

\item i.e., as $x\rightarrow x+\mu$ only shifts $K$ by additive
constant $ik\mu$, we only find ``central'' moments for coefficients of
$k^2$ and above.
\end{itemize}
\end{slide}

