

\section{\bf 5:\ \ Goodness of Fit}

\subsection{\bf 5.\hspace{1em} Goodness of Fit}
\subsection{\bf 5.1\hspace{1em} The $\chi^2$ Distribution}

\begin{slide}
\begin{itemize}
\item When the maximum likelihood method reduces to $\chi^2$, it is
relatively easy to judge goodness of fit.
\begin{equation}
\chi^2 = \sum^n_{i=1} \frac{(e_i - t_i)^2}{\sigma_i^2}
\end{equation}
where $t_i$ depend on $m$ theoretical parameters.  

\item Then the result of minimizing Equation~(33) with respect to $m$
theoretical parameters should be distributed according to $\chi^2$
distribution with $n-m$ degrees of freedom.

\item This function is plotted on p.~64 of EDJRS.  

\item If $k$ is number of degrees of freedom.
\begin{equation}
p(\chi^2) = \frac{\left( \frac{\chi^2}{2} \right)^{\frac{k}{2} - 1} \exp
\left( - \frac{\chi^2}{2} \right)}{2\Gamma \left(\frac{k}{2}\right)}
\end{equation}

\item $\langle\chi^2 \rangle = k$ and Standard Deviation is $\sqrt{2k}$.
\end{itemize}
\end{slide}

