Consider an experiment where two or three individuals are added to a population with probability $\lambda_{2}$ and $\lambda_{3}$ respectively per unit time. The death rate in the population is a constant $\beta$ per individual per unit time.

Write down the master equation for the probability $p_{n}(t)$ that there are $n$ individuals in the population at time $t$. From this, derive an equation for $\frac{\partial \phi}{\partial t}$, where $\phi$ is the generating function

$$\phi(s, t)=\sum_{n=0}^{\infty} s^{n} p_{n}(t)$$

Find the solution for $\phi$ in steady state, and show that the mean and variance of the population size are given by

$$\langle n\rangle=3 \frac{\lambda_{3}}{\beta}+2 \frac{\lambda_{2}}{\beta}, \quad \operatorname{var}(n)=6 \frac{\lambda_{3}}{\beta}+3 \frac{\lambda_{2}}{\beta} .$$

Hence show that, for a free choice of $\lambda_{2}$ and $\lambda_{3}$ subject to a given target mean, the experimenter can minimise the variance by only adding two individuals at a time.