

\subsection{\bf General $\chi^2$ Distribution}

\begin{slide}
\begin{itemize}
\item One can derive the $\chi^2$ distribution for $k$ degrees of
freedom in two ways.

\item 1.)\ Mathews and Walker 

\item Direct calculation as for $k=1$ noting that
${\displaystyle\sum^k_{i=1}} x^2_i$ with $x_i$ of mean 0, Standard
Deviation 1 is $r^2$ where $r$ is radial distance in $k$-dimensional
space.

\item Then factors multiplying $\exp \left( - \frac{\chi^2}{2} \right)$
in Equation~(34) are just volume element in $k$ dimensions.

\item 2.)\ Eadie, et al. show characteristic function of $\chi^2$ with one
degree of freedom is
\[
\int\limits^{+\infty}_{-\infty} \exp (i\,t\,\chi^2)\, p(\chi^2)\, d(\chi^2) =
\frac{1}{\sqrt{1-2\,i\,t}}
\]

\item Therefore, characteristic function for $\chi^2$ with $k$ degrees
of freedom is $(1-2\,i\,t)^{-\frac{k}{2}}$

\item and this can be inverse fourier transformed to give Equation~(34).
\end{itemize}
\end{slide}

