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\sect{Fluctuations in the number of particles}

A closed box of volume ${V_{0}}$ has ${N_{0}}$ particles. 
The "system" is a subvolume ${V}$. 
The number of particles in $V$ is a random variable ${N}$. 
Define the random variable ${\hat{X}_{n}}$, that indicates
weather the $n$th particle is located inside the system: 

\[
X_n=\left\{
\begin{matrix} 1, & \mbox{the particle is in } V \\
0, & \mbox{the particle is not in } V
\end{matrix}\right.
\]

\begin {itemize}

\item[(a)]
Express $\hat{N}$ using $\hat{X}_n$. Using theorems on adding independent random variables 
find $\langle N \rangle$ and $\mbox{Var}(N)$. 

\item[(b)]
Find the probability function ${f\left(N\right)}$  
using combinatorial considerations.
Calculate from it $\langle \hat{L}\rangle$ and $Var(L)$.

\item[(c)]Assume ${|(V/V_{0})-\frac{1}{2}| \ll 1}$, 
and treat ${N}$ as a continuous random variable. 
Apprximate the probability function ${f\left(N\right)}$ as a Gaussian, 
and verify agreemet with the central limit theorem.

\end {itemize}

