Past exam of the theoretical physics major option course of the University of Oxford / 2017 / 3 / c

For the Poisson process, $q_n=g$ and $r_n=0$, soThis is an ordinary differential equation in time for each fixed $z$, with solutionConsequentlyThe initial condition is interpreted as $P_0(0)=1$, with all other initial probabilities zero. Normalization follows from $F(1,t)=1$. The mean and variance are both $gt$, as expected for independent events arriving at constant rate.