Question:

Buses arrive at a bus station with i.i.d. interarrival times following an exponential distribution with intensity ?. Alice arrives at the bus station at a deterministic time t. a. (4pts) What is the expected waiting time for Alice until next bus comes? b. (6pts) Let ? be the time when the last bus arrived before time t. Show that t ? ? follows an exponential distribution with parameter ?. c. (6pts) Show that the expected interarrival time between the last bus which arrived before time t and the first bus which arrives after time t is 2 ? . Explain why it is different from the general expected interarrival time 1 ? .

4. Consider a discrete-time Markov chain with the following probability transition matrix 0 0 0 0 7 1-3-y P = 1-I-VVO T 0 0 1 0 Is it possible to choose values for a and y so that the Markov chain has the following properties? In each case, state the values of a and y, or give a brief reason why it is not possible. (a) The Markov chain has period 2. [2) (b) The Markov chain is reducible. (c) The Markov chain has at least one transient state. UNN (d) The Markov chain has invariant distribution (1/4, 1/4, 1/4, 1/4).public class RightTriangle ( public int hypotenuse () [ private int base; Teturn this -hypotenuse; private int height; private int hypotemine; public void getBase (int newBane; public RightTriangle this . bago . nowBang [int bane, ins height) get Hypotenuse ( ) this baoo - base, this, height - height, metHypotenuse !) ; public void sethleicht (Int newileig this. height . newHeight; getHypotenuse () : public int base () { return this . bases private void dotHypotenuse () this . hypotenuse = (int) Math rou public int height () Math. Bart (baga *bane + heightchoi return this . haight , / and RightTriangle