

\begin{slide}
\subsection{\bf $\chi^2$ Formulation of Linear Fit}
\subsection{\bf for Cross Section Data---I}

\begin{itemize}
\item Minimize: 
\[
{\displaystyle\sum^N_{i=1}} \frac{(x_i - \xi_i )^2}{\sigma^2_i}
\]
\begin{eqnarray*}
x_i & = & \mbox{const.}\ \ N_i \\
\sigma_i & = & \mbox{const.}\ \ \sqrt{N_i} \\
& = & \frac{x_i}{\sqrt{N_i}}
\end{eqnarray*}
where $i$'th bin (i.e., range of $\theta$ observed, which are centered
at $\theta_i$) has $N_i$ events.  

\item Minimizing $\chi^2$ with respect to unknown $a_m$ gives
\begin{eqnarray*}
\sum^N_{i=1} \frac{(x_i - \xi^*_i)}{\sigma^2_i}\ 
\frac{\partial\xi_i^*}{\partial a_m} & = & 0 \ \ \mbox{or} \\
&& \\
\sum_l M_{ml}\, a^*_l & = & b_m
\end{eqnarray*}

\item The superscript $^*$ denotes $\chi^2$ or maximum likelihood estimate.
\end{itemize}
\end{slide}

