

\subsection{\bf Particle Cluster Production II}

\begin{slide}
\begin{itemize}

\item To take a particularly simple case, consider a Poisson
distribution of clusters---each of which always contains four
particles.

\item For a random variable $x$ with a Poisson distribution
\begin{eqnarray*}
\mbox{Mean}\ = \lambda && \sigma^2 = \lambda \\
f_2 (x) = \sigma^2_x & - & \langle x \rangle = 0
\end{eqnarray*}

\item Now, let random variable $y=4x$
\begin{eqnarray*}
\langle y\rangle & = & 4\lambda \\
\left\langle y^2 \right\rangle & = & \left\langle (4x)^2 \right\rangle = 16
\left\langle x^2 \right\rangle \\
& = & 16 \left( \left\langle x^2 \right\rangle - \langle x\rangle^2 +
\langle x\rangle \right) \\
& = & 16 \left( \sigma_x^2 + \langle x\rangle^2 \right) \\
& = & 16 \left( \lambda + \lambda^2 \right) \\
f_2 (y) & = & \left\langle y^2 \right\rangle - \langle y\rangle^2 -
\langle y\rangle \\
& = & 16\left( \lambda + \lambda^2 \right) - 16\lambda^2 - 4\lambda \\
& = & 12\lambda > 0
\end{eqnarray*}

\item i.e., $y$ is always broader than a Poisson of mean $4\lambda$,
which has $f_2 = 0$. 
\end{itemize}
\end{slide}

