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\title{General Mathematics and ACM II}
\author{Exercise 11}
\date{March 23, 2011}
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\begin{enumerate}

\item Show that 
\[
  \sum_{k=0}^n \binom{n}{k} \, k \, x^k \, y^{n-k} 
  = n \, x \, (x+y)^{n-1} 
\]
and
\[
  \sum_{k=0}^n \binom{n}{k} \, k^2 \, x^k \, y^{n-k} 
  = n \, x \, (x+y)^{n-1} + n \, (n-1) \, x^2 \, (x+y)^{n-2} \,.
\]

\emph{Hint:} Differentiate the binomial theorem.

\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 the previous question.

\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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