Suppose buses arrive at a bus stop according to an approximate Poisson process at a mean rate of 5 per hour. Let Y denote the waiting time in minutes until the first bus arrives.

(a) How is Y distributed?

(b) What is the probability density function of Y.

(c) Suppose you arrive at the bus stop. What is the probability that you have to wait more than 5 minutes for the first bus?

(d) You take the bus every day this week. What is the probability that you have to wait less than 5 minutes for the first bus for exactly 3 of these days?

(e) Suppose 10 minutes have already passed since you started waiting for the first bus. What is the probability that the first bus will arrive after 15 minutes since you began waiting?