

\subsection{\bf (i) Example from Histogramming}
\subsection{\bf (Physics) Events---II}

\begin{slide}
\begin{itemize}
\item Here, we have a hierarchical situation, where we have a total
$N$ events of which $N_{\rm hist}$ fall in the particular histogram.
\begin{eqnarray*}
\frac{F}{G} & = & \frac{{\displaystyle \sum^N_{i=1}}\, \theta_i\,
g_i}{\displaystyle \sum^N_{i=1}\, g_i} \\
& = & \frac{\mbox{Number of Events in Bin}}{\mbox{Number of Events in
Histogram}}
\end{eqnarray*}

\item This estimate of fraction of events in histogram has standard
deviation
\[
\frac{\langle f\rangle}{\langle g\rangle}\ \sqrt{\frac{1}{N} \left\langle 
\left( \frac{f}{\langle f\rangle} - \frac{g}{\langle g\rangle} \right)^2
\right\rangle}
\]
\begin{eqnarray*}
& = & p\ \sqrt{\frac{1}{N\, \langle f\rangle^2}\ \left\langle 
(g\theta - g\,p)^2 \right\rangle} \\
& = & p\ \sqrt{\frac{\langle g^2 \rangle}{N\langle f\rangle^2}
\left\langle (\theta - p)^2 \right\rangle}
\end{eqnarray*}

\item $\theta$ is binomial and $\langle\theta^2 \rangle =
\langle\theta\rangle = p$

\item $\langle (\theta - p)^2 \rangle = p(1-p)$
\end{itemize}
\end{slide}

