Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 3 / 26J / ii

To derive the distribution from the infinitesimal characterization, let $p_j(t)=\mathbb{P}(N(t)=j)$. Independent increments give the forward equationsWith $L(t)={\int }_0^t\lambda (u)du$ and $p_0(0)=1$, $p_j(0)=0$ for $j>0$, these have solutionSubstitution verifies each equation, and the probabilities sum to one. Alternatively the probability generating function solves ${\partial }_{t}G=\lambda (t)(z-1)G$, $G(z,0)=1$, giving $G=e^{(z-1)L(t)}$. For a general locally integrable rate, time-changing a unit-rate Poisson process by $L(t)$ gives the same formula.