

\subsection{\bf Unnormalized Counter Experiment---I}

\begin{slide}
\begin{itemize}
\item Suppose the time $t$---which includes beam intensity, etc.---is
unknown.

\item Then we use Equation~(22) to determine it (regard $t$ as a
theoretical parameter)
\begin{eqnarray*}
\frac{\partial (\ln L)}{\partial t} & = & 0 \hspace{2em} \mbox{or} \\
t & = & \frac{{\displaystyle\sum^M_{i=1}} n_i}{{\displaystyle\sum^M_{i=1}}
\lambda_i} \\
&& \\
& = & \frac{N}{\Lambda}
\end{eqnarray*}
where $N = $ total Poisson parameter summed over all counters, and
$\Lambda = {\displaystyle\sum^M_{i=1}} \lambda_i$.

\item Now if we eliminate $t$, we get
\begin{eqnarray*}
\ln L & = & - \sum^M_{i=1} \ln n_i! - N + N \ln N \\
& + & \sum^M_{i=1} n_i \ln \left( \frac{\lambda_i}{\Lambda} \right)
\end{eqnarray*}
\end{itemize}
\end{slide}

