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\title{General Mathematics and CPS II}
\author{Exercise 11}
\date{March 21, 2012}
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\item Kac ring paper, Exercise 5.

\item  Consider an ensemble of Kac rings with $N$ sites where each site
is colored black independently with probability $\nu$.  Show that
\[
  \langle \Delta \rangle = (2\nu-1) \, N 
\]
and 
\[
  \operatorname{Var}[\Delta] = 4 \, \nu \, (1-\nu) \, N \,.
\]

\emph{Hint:} Use the result of Question 4 on the last exercise sheet.

\emph{Remark:} This is in fact a direct consequence of a well known
result on the variance of the binomial distribution which has a rather
elegant proof in terms of \emph{moment generating functions}.  These
are usually introduced in a first course on probability theory.  You
will likely encounter them this semester in Engineering and Science
Mathematics 2A/B.



\end{enumerate}
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