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 ? .

2. (15 pts) Consider a Markov chain { Xn } with state space S = {0, 1, 2} and transition matrix and transition matrix P = O ON/H HNIH O (1) Let the mapping f : S - S satisfy f(0) = 0 and f(2) = 1 and assume that f(1) + f(2). If Yn = f(Xn), then when is { Yn } a Markov chain? Is { Yn } always a Markov chain? In other words, are functions of Markov chains always Markov chains?1. Consider the Markov chain with the following transition matrix. 0 0.5 0.5 0.5 0 0.5 0.5 0.5 0 (a) Draw the transition diagram of the Markov chain. (b) Is the Markov chain ergodic? Give a reason for your answer. (c) Compute the two step transition matrix of the Markov chain. (d) What is the state distribution *2 for t = 2 if the initial state distribution for t = 0 is no = (0.1, 0.5, 0.4) ?2. Recall that Markov's inequality says that if T' is a positive-valued random variable with mean E(T) then E(T) P(T > 1 ) 0 there holds P(1X - #| Ska) 21 - 1 That is. with probability at least 1 - 2. X stays within & standard deviations around its mean