[QUE/SM-02004] SM-PROBLEM

There are $N$ ideal gas molecules in a room of volume $V=10\ {\rm M}^3$ and at room temperature. The molecules are in equilibrium. There is an imaginary box of volume $v\ {\rm M}^3$ completely contained within the room. The walls of the box are permeable and conducting. In other words, the box exchanges energy and matter with its  surroundings. Let $\nu$ denote a random variable defined as the number of molecules in the box. Let $P(n)$ be the probability that $\nu=n$. Let $\mu=Nv/V$. Derive appropriate expressions for  $P(n)$ to describe the statistics of $n$ for the following cases :

Calculate the mean and variance of the random variable $\nu $ for both cases.