A large population of some species has probability $P(n, t)$ of taking the value $n$ at time $t$. Explain the use of the generating function $\phi(s, t)=\sum_{n=0}^{\infty} s^{n} P(n, t)$, and give expressions for $P(n, t)$ and $\langle n\rangle$ in terms of $\phi$.

A particular population is subject to a birth-death process, so that the probability of an increase from $n$ to $n+1$ in unit time is $\alpha+\beta n$, while the probability of a decrease from $n$ to $n-1$ is $\gamma n$, with $\gamma>\beta$. Show that the master equation for $P(n, t)$ is

$$\frac{\partial P(n, t)}{\partial t}=(\alpha+\beta(n-1)) P(n-1, t)+\gamma(n+1) P(n+1, t)-(\alpha+(\beta+\gamma) n) P(n, t)$$

Derive the equation satisfied by $\phi$, and show that in the statistically steady state, when $\phi$ and $P$ are independent of time, $\phi$ takes the form

$$\phi(s)=\left(\frac{\gamma-\beta}{\gamma-\beta s}\right)^{\alpha / \beta}$$

Using the equation for $\phi$, or otherwise, find $\langle n\rangle$.