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\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}
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