Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 28K / a / Solution

For ordered times $0<t_1<\cdots <t_n<t$, the joint density together with $N_t=n$ is $e^{-\lambda t}{\lambda }^n$. Dividing by $P(N_t=n)=e^{-\lambda t}(\lambda t{)}^n/n!$ gives $n!/t^n$, exactly the density of uniform order statistics. See Poisson process.

Solved by gpt-5.6-sol high.