

\subsection{\bf Derivation of $\chi^2$ from}
\subsection{\bf Maximum Likelihood}

\begin{slide}
\begin{itemize}
\item Then the probability of each observation is
\[
p_i = \frac{1}{\sqrt{2\pi} \sigma_i} \exp \left\lbrace \frac{- (x_i -
\xi_i )^2}{2\sigma^2} \right\rbrace
\]

\item The likelihood
\[
L = \prod^N_{i=1} p_i \propto \exp \left\lbrace - \sum^N_{i=1}
\frac{(x_i - \xi_i)^2}{2\sigma^2_i} \right\rbrace
\]

\item so, the principle of maximizing the likelihood is equivalent to
minimizing $\chi^2$ where
\begin{equation}
\chi^2 = \sum^N_{i=1} \frac{(x_i - \xi_i)^2}{\sigma^2_i}
\end{equation}

\item with respect to $n_T$ theoretical parameters in $\xi_i$.
\end{itemize}
\end{slide}

