

\subsection{\bf Method of Moments---III}

\begin{slide}
\begin{itemize}
\item (c) Error in $\mu$ is theoretically
\[
\frac{1}{\sqrt{N}}\, \sqrt{\int\limits f^2 p(x, \alpha) dx - \mu^2_0}
\]
\begin{description}
\item{$\bullet$} and experimentally
\[
\frac{1}{\sqrt{N}}\, \sqrt{\frac{1}{N} \sum^N_{i=1} f^2 (x_i) - \left(
\frac{1}{N} \sum^N_{i=1} f(x_i) \right)^2}
\]
\item{$\bullet$} using $\mu_0$ to be true value of mean and 
\item{$\bullet$} $\mu = \frac{1}{N} {\displaystyle\sum^N_{i=1}} f(x_i)$
to be estimate.
\end{description}

\item By central limit theorem, for large $N$, experimental estimate
will give correct error.

\item However, if all $x_i$ are the same (always true if $N=1!$), then
experimental error = 0, which is simply wrong.

\item The moral is: use theoretical estimate if you only have a few
events.
\end{itemize}
\end{slide}

