%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\sect{Average length of a polymer}

A polymer can be described as a chain of ${N}$ monomers. 
Each monomer has the probability ${p}$ to be positioned horizontally, 
adding length ${a}$ to the polymer, 
otherwise the monomer adds length ${b}$. 
Let ${L}$ be the total length of the polymer. 
Define random variables $\hat{X}_n$ such that:

\[
X_n=
\left\{\begin{matrix} 
a, & \mbox{the monomer is horizontal} \\
b, & \mbox{the monomer is vertical}
\end{matrix}\right.
\]

\begin {itemize}

\item[(a)]
Express $\hat{L}$ using $\hat{X}_n$. 
Using theorems for adding independent random variables 
find the average length $\langle L \rangle$ 
and the variance $\mbox{Var}(L)$. 

\item[(b)]
Define  $f\left(L\right)\equiv {P}\left(L=na+\left(N-n\right)b\right)$. 
Find it using combinatorial considerations. 
Calculate $\langle \hat{L}\rangle$ and $\mbox{Var}(L)$.

\item[(c)]
Define $\sigma_{L} = \sqrt{\mbox{Var}(L)}$. 
What is the behavior of ${\sigma_{L}/\langle L\rangle}$ as a function of ${N}$?

\end {itemize}

%%begin{figure}
\putgraph{Ex002}
%%\caption{}\label{}
%%end{figure}




