Suppose a bus arrives at a station such that the time between arrivals is exponentially distributed with rate 1/λ. to get home, you decide to wait for the bus for some number of minutes t. if the bus has arrived before t minutes, you take the bus home which takes time b. if the bus has not arrived after t minutes, you walk home which takes time w.

(a) what is the expected total time from getting to the bus stop until getting home?

(b) suppose w < 1/λ + b at what value of t is the expected wait time minimized?

(c) suppose w > 1/λ + b at what value of t is the expected wait time minimized?