# Suppose a bus arrives at a station such that the time between arrivals is exponentially distributed with rate 1/. To get home, you decide to wait for…

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?