Question 1 1.1 Exercise 4.24 [ITPM-ROSS] 1.2 A Markov chain {Xn, n 0} with states 0, 1, 2, 3, 4 has the transition matrix: P = [1/4 0 1/4 1/4 1/4 0 0 1/2 1/4 0 0 1 1/4 1/4 1/4 1/2 1/4 1/4 0 1/4 0 Question 3 0 0 1/4 Specify/identify the classes of this Markov chain, and determine whether they are transient or recurrent. Question 2 2.1 Exercise 5.5 [ITPM-ROSS] 2.2 Under which conditions is a counting process N(t) with rate a considered to be a Poisson Process? 2.3 Exercise 5.34 [ITPM-ROSS] 3.1 Describe what a Brownian process B (t) is. 3.2 Exercise 10.6 [ITPM-ROSS] 3.3 In a bicycle race between two competitors, let Y(t) denote the amount of time (in seconds) by which the racer that started in the inside position is ahead when 100t percent of the race has been completed, and suppose that {Y(t), 0 t 1} can be effectively modeled as a Brownian motion process with variance parameter o. If the inside racer is leading by o seconds at the midpoint of the race, what is the probability that he is the winner?