Customers and taxis arrive at a taxi stand according to two independent Poisson processes with respective rates and . As soon as a taxi and a customer are available, they leave the stand. Suppose that at time 0 the stand is empty, i.e., there are no taxis or customers.

(a) Let N(t) denote the number of arrivals of both taxis and customers. What is the probability that the first arrival of N is a taxi?

(b) Let N₁(t) denote the number of arrivals of taxis. Find E(N₁(t) N(t + s)).

(c) Let X denote the number of customers arriving before the first arrival of taxis. Find the probability mass function of X.

(d) Let T be the time of the first departure of a taxi-customer pair. What is the expected value of T?