

\subsection{\bf Determination of Lifetime Example}
\subsection{\bf of Maximum Likelihood---I}

\begin{slide}
\begin{itemize}
\item All books give the example of determining the lifetime $\tau_0$
of a particle from the observation of decay times $t_1 \ldots t_n$ of
$n$ decays.

\item The probability distribution for decay problem is Poisson:
\[
p(t) dt = \frac{1}{\tau} \exp \left( - \frac{t}{\tau} \right) dt
\]

\item Now, $dt$ is meaningless---it is independent of $\tau$ and only
affects normalization of likelihood.

\begin{description}
\item{$\bullet$} The unknown $P(\alpha\vert T)$, $P(x_i \vert T)$ terms
in Equation~(14) made this arbitrary anyway.
\end{description}

\item We must, however, keep terms in normalization $\left(
\frac{1}{\tau}\ \mbox{here} \right)$ that depend on parameters to be
determined.

\item The likelihood of $n$ decays is:
\begin{eqnarray*}
L(\tau) & \propto & \frac{1}{\tau}\, e^{-\frac{t_1}{\tau}}\ \ldots\ 
\frac{1}{\tau}\, e^{-\frac{t_n}{\tau}} , \hspace{1em} \mbox{or} \\
L(\tau) & = & \exp \left\lbrack - \frac{1}{\tau}\, {\displaystyle \sum^n_{i=1}}
t_i - n \ln \tau \right\rbrack
\end{eqnarray*}
where we lazily convert $\propto$ into an = sign.  
\end{itemize}
\end{slide}

