Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 28L (Applied Probability)

Let $p_n(t)=\mathbb{P}(X_t=n)$. The infinitesimal definition gives the forward equationswith $p_0(0)=1$. Solving recursively givesso $X_t\sim \mathrm{Poisson}(\lambda t)$.

Conditionally on $X_t=n$, the $n$ arrival times are the order statistics of independent $\mathrm{Unif}[0,t]$ variables. For $n\ge 1$, the last jump occurs before $3t/4$ exactly when every point lies before that time, so the probability isThe number falling in $(t/4,3t/4)$ is therefore

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