Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 2 / 25K (Applied Probability)

A rate-$\lambda$ Poisson process has $N_0=0$, right-continuous paths with upward jumps of size one, stationary independent increments, and $N_t-N_s\sim \mathrm{Poisson}(\lambda (t-s))$ for $t\ge s$. Assume $\lambda >0$. Its successive waiting times are independent exponential variables of rate $\lambda$. For $0<t_1<\cdots <t_n<t$, the product of their densities and the probability of no further jump before $t$ isDivide by $\mathbb{P}(N_t=n)=e^{-\lambda t}(\lambda t{)}^n/n!$ to obtain the conditional joint densityBoundary inequalities do not affect a density. Equivalently these are the ordered values of $n$ independent uniform points of $[0,t]$. For $n=0$, the conditional tuple is empty.

Assume the zero set $A=\{s\in [0,t]:g(s)=0\}$ is measurable, as is ensured by measurability of $g$. This is an implicit necessary hypothesis: for an arbitrary real function the printed integral and event need not be measurable. Put $a={\int }_0^t{1}_A(s)ds$. Conditional on $N_t=n$, the uniform order statistics description makes the probability of no jump in $A$ equal to $(1-a/t{)}^n$. Averaging over the Poisson distribution givesConsequentlyIn particular a measure-zero zero set is almost surely missed, despite possibly containing infinitely many points.