Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 3 / 13A / a

This birth-death process has birth rates $b_n=\alpha +\beta n$ and death rates $d_n=\gamma n(n-1)$. With $P_{-1}=0$, its birth-death master equation isThe first two terms are inflow by a birth from $n-1$ and a death from $n+1$; the bracketed term is the total outflow from $n$.