
\subsection{\bf 1.7\hspace{1em}  Leading Up to the Central}
\subsection{\bf Limit Theorem Moment}
\subsection{\bf Generating Function---I}

\begin{slide}
\begin{itemize}
\item The Moment Generating Function which is also called
characteristic function, is defined in terms of probability
distribution $p(x)$ by
\begin{eqnarray*}
\varphi (k) & = & \int\limits^{+\infty}_{-\infty} dx\, e^{ikx} p(x) \\
& = & 1 + ik\langle x\rangle - \frac{k^2}{2!} \left\langle x^2 \right\rangle
+ \ldots
\end{eqnarray*}

\item Note the Fourier inversion theorem gives
\[
p(x) = \frac{1}{2\pi} \int\limits^{+\infty}_{-\infty} dx\, e^{-ikx} \varphi (k)
\]
\end{itemize}
\end{slide}

