

\subsection{\bf Solution and Errors from Linear $\chi^2$}

\begin{slide}
\begin{itemize}
\item We can write the result in matrix form
\[
\mbox{\bf a}^* = \mbox{\bf M}^{-1} \mbox{\bf b}
\]

\item We can find errors in either of the two ways
\begin{description}
\item{$\bullet$} a) regard $\mbox{\bf a}^*$ as a function of $x_i$ and
calculate standard deviation from central limit theorem.

\item{$\bullet$} b) Use the shape of likelihood, $L=\exp \left( -
\frac{1}{2} \chi^2 \right)$.
\end{description}

\item We have---as far as shape of $L$ is concerned:

$L = \exp$ (independent of $\mbox{\bf a}+$ no terms linear in
$(\mbox{\bf a} -\mbox{\bf a}^*)$ due to ``maximum'' condition
$-\frac{1}{2} (\mbox{\bf a} - \mbox{\bf a}^*)^T \mbox{\bf M} (\mbox{\bf a} -
\mbox{\bf a}^*)$),

\item i.e., $\chi^2$ matrix {\bf M} is indeed error matrix for $a$.
\end{itemize}
\end{slide}

