Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 28K / b / i / Solution

Starting with one individual, let $S$ be the first event time, of density
$(\lambda +\mu )e^{-(\lambda +\mu )s}$. At that event there is a death with conditional probability $\mu /(\lambda +\mu )$, causing extinction immediately, or a birth with probability $\lambda /(\lambda +\mu )$, leaving two independent descendant populations. The branching property therefore gives