

\subsection{\bf Example of Moment Method}

\begin{slide}
\begin{itemize}
\item Consider
\[
p(x) = \frac{1}{2} (1 + ax) \hspace{2em} -1 \le x \le 1
\]
\item Use Moment Method for any power $m$ of $x$
\begin{eqnarray*}
\langle x^m \rangle & = & \frac{a}{2} \int\limits^{+1}_{-1} x^{m+1} dx \\
& = & \frac{1}{m+2} \hspace{2em} : m\ \ \mbox{odd}
\end{eqnarray*}

\item So if $M_m = $ moment method using $\langle x^m \rangle$ and let
$M_m$ produce estimate $a^*_m$.

\item Then
\[
a^*_m = (m+2) \sum^N_{i=1} \frac{x^m_i}{N}
\]
\item The $\rm error^2$ in estimate $a^*_m$ is
\begin{eqnarray*}
\sigma^2 \big\vert_m & = & \frac{(m+2)^2}{N} \left\lbrace
\int\limits^{+1}_{-1} x^{2m} (1+ax) dx - \frac{a^2}{(m+2)^2} \right\rbrace \\
& = & \frac{1}{N} \left\lbrace \frac{(m+2)^2}{2m+1} - a^2
\right\rbrace
\end{eqnarray*}

\item which is a minimum for $m=1$ when
\[
\sigma^2 \big\vert_1 = \frac{1}{N} (3-a^2)
\]
\end{itemize}
\end{slide}

