Suppose that people arrive at a bus stop in accordance with a Poisson process with rate λ. The bus departs at time t. Let X denote the total amount of waiting time of all those who get on the bus at time t. We want to determine Var(X). Let N(t) denote the number of arrivals by time t.

(a) What is E[X|N(t)]?

(b) Argue that Var[X|N(t)] = N(t)t2/12.

(c) What is Var(X)?