

\subsection{\bf $\chi^2$ Distribution for One}
\subsection{\bf Degree of Freedom}

\begin{slide}
\begin{itemize}
\item Consider $\chi^2$ with one degree of freedom
\begin{eqnarray*}
\chi^2 & = & \frac{(x-t)^2}{\sigma^2} \ \mbox{where} \\
p(x) & = & \frac{1}{\sigma\sqrt{2\pi}} \exp \left( -
\frac{(x-t)^2}{2\sigma^2} \right) \\
p(\chi^2 ) & = & p(x) \frac{dx}{d\chi^2} = \frac{p(x)\sigma^2}{2(x-t)}
\cdot 2 \\
p(\chi^2 ) & = & \frac{\exp \left( - \frac{\chi^2}{2} \right)}{\sqrt{2\pi}\ 2\
\sqrt{\chi^2}} \cdot 2
\end{eqnarray*}

\item which is value in Equation~(34) using $\Gamma (1/2) = \sqrt{\pi}$ 

\item (Here, factor of two comes as $\pm (x-t)$ given same $\chi^2$ value).
\end{itemize}
\end{slide}

