Buses arrive at a stop according to a Poisson process (N_t)_t0 with rate . The arrival times are denoted by (S_n)_n1. Every morning you arrive at the bus stop at a fixed time t > 0. Therefore, S_Nt is the time when the last bus passes before your arrival at the stop, and S_Nt+1 will be the time when you board the next bus.

You have a neighbour who is a bit magical: each day he manages to catch the bus just before yours.

(a) Argue that V_t = S_Nt+1 t, your waiting time for your bus, follows an exponential distribution with rate parameter .

(b) (i) By conditioning on N_t , find the cumulative distribution function of S_Nt , the departure time of your neighbour.

(ii) Show that E(S_Nt ) = t 1/ + 1 /( e ^t) .

(c) Calculate E(S_Nt+1 S_Nt ), the average time your neighbour departs ahead of you.