Past exam of the probability, statistics and options I course of the University of Oxford / 2008 / M / 3 / Solution

A Poisson process of rate $\lambda >0$ starts from zero, has independent stationary increments, has unit jumps, and satisfies $N_t-N_s\sim \mathrm{Poisson}(\lambda (t-s))$ for $t>s$. Its inter-arrival times are independent with the exponential distribution of rate $\lambda$: any finite joint density is ${\prod }_i\lambda e^{-\lambda x_i}$ on positive coordinates.

The second arrival is a sum of two such variables. Convolution givesThis is a gamma distribution of shape two and rate $\lambda$. If $S$ is the first arrival of the independent rate-$\mu$ process, exponential independence gives