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)?