Past exam of the mathematics course of the University of Cambridge / 2018 / ii / Paper 1 / 6C / b / Solution

Multiply the birth-death master equation by $n$ and sum. Index shifts giveFor the second moment, a birth changes $n^2$ by $2n+1$ and a death changes it by $-2n+1$, soSince ${\sigma }^2=\mathbb{E}[N^2]-{\mu }^2$,Together these are the moment equations of a linear birth-death process.