

\subsection{\bf The Likelihood Maximizes the}
\subsection{\bf Quivering Rod---I}

\begin{slide}
\begin{itemize}
\item Consider, again, the explicit form of the likelihood of a rod
quivering in the earthquake.
\[
L\left( x_1 \ldots x_n \right) = c \exp \left\lbrace -
\frac{1}{2\sigma^2} \sum^n_{i=1} \left( x_i - \alpha \right)
\right\rbrace^2
\]

\item According to Equation~(14), we now regard $L$ as a function of
$\alpha$ and so rewrite it:
\begin{eqnarray*}
L(\alpha) & = & c\, \exp \left\lbrace - \frac{n}{2\sigma^2} \left(
\alpha - \frac{1}{n}\ {\displaystyle \sum^n_{i=1}} x_i \right)^2 \right. \\
& - & \frac{1}{2\sigma^2} \left\lbrack \sum_i x_i^2 - \frac{1}{n}
\left.
\left( \Sigma_i x_i \right)^2 \right\rbrack\right\rbrace 
\end{eqnarray*}

\item On completing the square to write exponent as quadratic in
$\alpha$.

\end{itemize}
\end{slide}

