

\subsection{\bf Derivative Formalism for $\chi^2$---II}

\begin{slide}
\begin{itemize}
\item Now, $S_1$ is $O(\sqrt{N})$ and asymptotically negligible.

\item To show this, regard it as a random
variable which is a function of the $N$ random variables $e_1 \ldots
e_N$.  

\item By the Central Limit Theorem, $S_1$ has mean
\[
\mu = \sum^N_{i=1} \mu_i
\]

\item and Standard Deviation $\sigma$.
\[
\sigma^2 = \sum^N_{i=1} \omega_i^2
\]

\item where
\[
\frac{2(t_i - e_i)}{\sigma^2_i}\ \frac{\partial^2 t_i}{\partial\alpha^2}
\]
has mean $\mu_i$ and Standard Deviation $\omega_i$.  
\end{itemize}
\end{slide}

