Answer Happy

You are waiting on the platform of the first stop of a Manhattan subway line. You could ride either a local or express train to your destination, which is the last stop on the line. The waiting time X for the next express train is the exponential random variable with E[X] = 10 minutes. The waiting time Y for the next local train is the exponential random variable with E[Y] = 5 minutes. Although the arrival times X and Y of the trains are random and independent, the trains' travel times are deterministic; the local train travels from first stop to last stop in exactly 15 minutes while the express travels from first to last stop in exactly 5 minutes. (a) What is the joint PDF fx,y(x, y)? (b) Find P[L] that the local train arrives first at the platform? (C) Suppose you board the first train that arrives. Find the PDF of your waiting time W = min(X,Y). (d) The time until the first train (express or local) reaches final stop is T = min(X + 5, Y + 15). Find fr(t). (e) Suppose the local train does arrive first at your platform. Should you board the local train? Justify your answer. (There may be more than one correct answer.)