Consider a birth-death process in which the birth and death rates in a population of size $n$ are, respectively, $B n$ and $D n$, where $B$ and $D$ are per capita birth and death rates.

(a) Write down the master equation for the probability, $p_{n}(t)$, of the population having size $n$ at time $t$.

(b) Obtain the differential equations for the rates of change of the mean $\mu(t)=\langle n\rangle$ and the variance $\sigma^{2}(t)=\left\langle n^{2}\right\rangle-\langle n\rangle^{2}$ in terms of $\mu, \sigma, B$ and $D$.

(c) Compare the equations obtained above with the deterministic description of the evolution of the population size, $d n / d t=(B-D) n$. Comment on why $B$ and $D$ cannot be uniquely deduced from the deterministic model but can be deduced from the stochastic description.