

\begin{slide}
\subsection{\bf Mean and Standard Deviation of $\chi^2$}
\begin{itemize}
\item Notice that $\chi^2$ is the sum ${\displaystyle \sum^N_{i=1}}
v_i^2$ where $v_i$ are independent Gaussian random variables of unit
standard deviation and zero mean.  

\item We can calculate the $\chi^2$ distribution exactly (Mathews and
Walker pages 393--395) but in practice this is rarely necessary.

\item Thus, we can easily show that---for fixed $\xi_i$---that
\begin{equation}
\langle\chi^2 \rangle = N \hspace{3em} \sigma^2_{\chi^2} = 2N
\end{equation}
and this is sufficient for large $N$ as central limit theorem says
$\chi^2$ will be Gaussian with above mean and standard deviation.

\item Now Equation~(20) is not quite correct because if we determine
the theoretical parameters to minimize $\chi^2$, then implicitly
$\xi_i$ are also random variables that are functions of $x_j$.

\item It is not too hard to show that in this case one should just
replace $N$ by $N-n_T$ (number of degrees of freedom) in (20).

\item In any case, (20) is very important because it allows an easy
criterion for the goodness of fit.
\end{itemize}
\end{slide}

