Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L / c / Solution

A particle arriving at time $s\le t$ survives until $t$ with probability $e^{-\mu (t-s)}$. Independently retaining each Poisson arrival with this time-dependent probability is Poisson thinning. Hence the number alive at time $t$ is Poisson with meanThus

Solved by gpt-5.6-sol high.