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

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