At a subway station, eastbound trains and northbound trains arrive independently, both according to a Poisson process. On average, there is one eastbound train every 12 minutes and one northbound train every 8 minutes. Suppose you arrive at the subway station at a certain point in time and start observing trains.
(a) What is the probability that no eastbound trains arrive between during the first 30 minutes you are observing the trains?
(b) How many eastbound trains are expected during a 30 minute period?
(c) Assuming you arrived at 8:00, at what time can you expect the 5th eastbound train?
(d) What is the probability that exactly 2 eastbound trains will arrive in the first 24 minutes and exactly 3 northbound trains will arrive in the first 36 minutes?
(e) What is the expected waiting time, in minutes until the first train (of either type) arrives?