

\subsection{\bf $\chi^2$ Characteristic Function}

\begin{slide}
\begin{itemize}
\item Introduce the Characteristic Function $F(\chi^2)$ by
\begin{eqnarray*}
F(\chi^2) & = & \int\limits^{\chi^2}_0 p(\chi^2) d(\chi^2) \\
& = & \frac{1}{2\Gamma \left(\frac{k}{2}\right)} \int\limits^{\chi^2}_0
\left(\frac{u}{2}\right)^{\left(\frac{k}{2}\right) - 1}
e^{\frac{-u}{2}} du
\end{eqnarray*}

\item $F$ has probability distribution $w(F)$
\[
w(F) = p(\chi^2) \frac{d(\chi^2)}{dF} = 1
\]
so $F$ is uniformly distributed between 0 and 1.
\end{itemize}
\end{slide}

