A dolphin spends its time either feeding out at sea (state 0), or resting in coastal waters (state 1). Each day it chooses one of these two states. Let {X1, X2, X3, ...} be the state of the dolphin on days 1, 2, 3, . . ., where each = {0,1}. The dolphin's behaviour is modelled by a Markov chain following the transition diagram below. 1/2 1/2 3/4 0 1 Sea Coast 1/4 a)(20 pts) Write down the transition matrix, P, for the Markov chain represented by the above diagram. b)(30 pts) Suppose the dolphin is seen at the coast on day 1. Find the probability it will remain at the coast on day 2, and go out to sea on day 3. c)(30 pts) Show that = () is a stationary distribution for this Markov chain. d)(20 pts) Does the Markov chain converge to the stationary distribution as t o? Explain why or why not.