

\subsection{\bf Normalized Counter Experiment}

\begin{slide}
\begin{itemize}
\item Now, we suppose $t$ is no longer an unknown theoretical
parameter, but a known number.
\end{itemize}
\[
\ln L = \mbox{const.}\ + \underbrace{\lbrace N \ln (t\Lambda) -
t\Lambda \rbrace}_{(a)} + \underbrace{\sum^M_{i=1} n_i \ln
(\lambda/\Lambda)}_{(b)}
\]
\begin{itemize}
\item (b) is the form we found before. 

\item (a) can be regarded as $\ln p_N$ where $p_N$ is probability of
getting $N$ (which should now be regarded as an experimental
observable) total counts in time $t$

\item ($p_N = e^{-t\Lambda} (t\Lambda)^N$ which is up to a constant,
the usual Poisson distribution).

\item This requires some care to apply previous theorems as
probabilities must be normalized in proofs in Equation~(14).  However,
theorems do hold.

\item Note Orear calls this normalized case as the extended maximum
likelihood method.

\end{itemize}
\end{slide}

