Provide work

3. (20 points) A certain stochastic process is modeled by a Markov chain where the state variable is Xt, and the state space is S = {1, 2, 3, 4, 5}. The index set is T = {0, 1, 2, 3, 4, 5, 6}, and t represents time in hours. The transition matrix is given below. 0 0.4 0 0.4 0.2 0.05 0.45 0.4 0 0.1 P = 0.4 0.4 0.2 0 0 0 0 0 1 0.6 0.05 0.25 (a) What is P{ X4 = 5 X3 = 1}? (b) If the process is initially in state 3 at time 0, is it possible to get state 4 at T = 2? If yes, give the realization and calculate its probability; If no, state the reason. (c) If the process is initially in state 2 at time 0, what is the probability that the process goes to state 3, state 1 and returned to state 2 sequentially? (d) Draw a transition diagram for the Markov chain