7 Waiting For the Bus

Edward and Jerry are waiting at the bus stop outside of Soda Hall.

Like many bus systems, buses arrive in periodic intervals. However, the Berkeley bus system is unreliable, so the length of these intervals are random, and follow Exponential distributions.

Edward is waiting for the 51B, which arrives according to an Exponential distribution with parameter . That is, if we let the random variable X_i correspond to the difference between the arrival time ith and i1st bus (also known as the inter-arrival time) of the 51B, X_i Expo().

Jerry is waiting for the 79, whose inter-arrival time also follows an Exponential distribution with parameter . That is, if we let Y_i denote the inter-arrival time of the 79, Y_i Expo(). Assume that all inter-arrival times are independent.

(a) What is the probability that Jerry's bus arrives before Edward's bus?

(b) After 20 minutes, the 79 arrives, and Jerry rides the bus. However, the 51B still hasn't arrived yet. Let D be the additional amount of time Edward needs to wait for the 51B to arrive. What is the distribution of D?

(c) Lavanya isn't picky, so she will wait until either the 51B or the 79 bus arrives. Solve for the distribution of Z, the amount of time Lavanya will wait before catching the bus. (d) Khalil arrives at the bus stop, but he doesn't feel like riding the bus with Edward. He decides that he will wait for the second arrival of the 51B to ride the bus. Find the distribution of T = X₁ +X₂, the amount of time that Khalil will wait to ride the bus.