

\subsection{\bf Derivative Calculation for}
\subsection{\bf Maximum Likelihood---II}

\begin{slide}
\begin{itemize}
\item Now, $S_{1,2}$ are directly of form suitable to apply central
limit theorem. 

\item Thus, they are sums over independent random variables
taken from same distribution.
\begin{eqnarray*}
\langle S_1 \rangle & = & N \left\langle \frac{1}{q}\, \frac{\partial^2
q}{\partial\alpha^2}\right\rangle \\
& = & N \int\limits q \cdot \frac{1}{q}\, \frac{\partial^2 q}{\partial\alpha^2}
d\alpha \\
& = & N \frac{\partial^2}{\partial\alpha^2} \left\lbrace \int\limits q\,
d\alpha \right\rbrace \\
& = & N \frac{\partial^2}{\partial\alpha^2} (1) = 0
\end{eqnarray*}

\item So again $S_1$ only has standard deviation term and is
$O(\sqrt{N})$. 

\item Similarly $S_2$ is $O(N)$ and dominates.  

\item It follows
that exactly same methods can be applied to $\chi^2$ and maximum
likelihood fits. 

\item In the future we will only mention the former.
\end{itemize}
\end{slide}

