

\subsection{\bf Individual Events Become}
\subsection{\bf Binned Data---I}

\begin{slide}
\begin{itemize}
\item If we have so many events that I can divide them into $K$ groups
of $n_k$ ($k=1\ldots K$) events where $p$ has the same value for each
member of the group, then:
\begin{eqnarray}
L & = & \exp \left\lbrace - \int\limits p\, dx\, t \right\rbrace \prod^K_{k=1}
(pt)^{n_k} \nonumber \\
& \propto & \exp \left\lbrace - \sum^K_{k=1} \lambda_k t \right\rbrace
\prod^K_{k=1} (\lambda_k t)^{n_k}
\end{eqnarray}
where 
\[
\lambda_k = \int_{\, \mbox{\small bin for k'th group}}\, p\, dx,
\]
and we return exactly to formulae in Section~3.3. 

\item Further, if $n_k$ is large for each $k$, our Poisson
distributions become Gaussian and we can use $\chi^2$ method to analyze
data.  

\begin{description}
\item{$\bullet$} This is the only case where $\chi^2$ is correct in the
sense of giving the best answer.
\end{description}
\end{itemize}
\end{slide}

