Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 3 / 13C / a / i / Solution

This is an immigration--death process. New individuals enter independently of the current population at the constant rate $\lambda$, producing transitions $n\to n+1$. Each existing individual dies at rate $\beta$, so the total transition rate $n\to n-1$ is $\beta n$.