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\sect{Fluctuations in the number of particles}

${N_{0}}$ particles in a closed tank with volume ${V_{0}}$ are
given. we'll focus on an area with volume ${V}$  which we'll pronoun
"The system". The number of the particles in the system is a random
variable ${N}$.

\begin {itemize}

\item[(a)]
Find ${\langle N\rangle}$ and ${\sigma_{N}}$  using  the probability
theory.

(guideline: define random variables ${\hat{X}_{n}}$ which determines
if a certain particle is in the system. i.e. ${X_{n}=0}$ or ${1}$).

\item[(b)]
Find the probability function ${f\left(N\right)}$.

\item[(c)]
Assume ${|V/V_{0}-\frac{1}{2}|<<1}$ and treat ${N}$ as a continuous
random variable. Find the probability function ${f\left(N\right)}$
in this assumptions framework.

\end {itemize}