

\subsection{\bf Error Estimate from $\chi^2$ Value}

\begin{slide}
\begin{itemize}
\item If we expand $\chi^2$ near its minimum with---for convenience---just
one theoretical parameter, $\lambda$.
\begin{equation}
\chi^2 = \chi^2_0 + \alpha \left( \lambda - \lambda_0 \right)^2
\end{equation}

\item where $\lambda_0$ is our estimate of $\lambda$.  

\item Then probability distribution of $\lambda$ is just
\begin{eqnarray*}
\alpha \exp \left( - \frac{\chi^2}{2} \right) \\
\alpha \exp \left( - \frac{\alpha(\lambda - \lambda_0 )^2}{2}\right)
\end{eqnarray*}
i.e., $\frac{1}{\sqrt{\alpha}}$ is Standard Deviation of estimate.

\item Returning to Equation~(35), we find that if we put $\lambda =
\lambda_0 \pm \frac{1}{\sqrt{\alpha}}$ (i.e., at its one standard
deviation value), 

\item $\chi^2$ is increased by 1.  

\item This $\chi^2$ increase ``rule'' is common method of finding
``error'' in estimate $\lambda$ even when the Taylor expansion equation
is not a good approximation.
\end{itemize}
\end{slide}

