

\subsection{\bf Bubble/Spectrometer---Physics}
\subsection{\bf Individual Events with Continuous}
\subsection{\bf Distributions---III}

\begin{slide}
\begin{itemize}
\item Let $N$ be total number of events:  then likelihood is
\begin{eqnarray*}
L & = & \prod_{\rm successes} p \mbox{(success)} \prod_{\rm failures}
p \mbox{(failure)} \\
& = & \prod^N_{i=1} \exp (-p_i\, t\, dx_i) (p_i\, dx_i\, t) \\
& \otimes & \prod^{M-N}_{j=1} \exp (-p_j\, dx_j\, t)
\end{eqnarray*}

\item writing out $L$ in terms of Poisson probabilities for 0 or 1
events.

\item Now $M>>N$ and $M$ bins $dx_j$ cover space.  Thus
\[
L = \exp \left\lbrack - {\displaystyle\int\limits} p(x)\, dx\, t \right\rbrack 
\prod^N_{i=1} p_i\, dx_i\, t
\]
\end{itemize}
\end{slide}

