Consider a birth and death process in which births always give rise to two offspring, with rate $\lambda$, while the death rate per individual is $\beta$.

Write down the master equation (or probability balance equation) for this system.

Show that the population mean is given by

$$\langle n\rangle=\frac{2 \lambda}{\beta}\left(1-e^{-\beta t}\right)+n_{0} e^{-\beta t}$$

where $n_{0}$ is the initial population mean, and that the population variance satisfies

$$\sigma^{2} \rightarrow 3 \lambda / \beta \quad \text { as } \quad t \rightarrow \infty$$