A discrete-time Markov chain with seven states has the following transition probabilities: pij = 0.5 , (i, j) = (3, 2), (3, 4), (5, 6) and (5, 7) 1 , (i, j) = (1, 3), (2, 1), (4, 5), (6, 7) and (7, 5) 0 , otherwise In the questions below, we let Xk be the state of the Markov process at time k. (a) Give a pictorial representation of the discrete-time Markov chain. (b) For what values of n is the probability r15(n) = P(Xn = 5 | X0 = 1) > 0? (c) What is the set of states A(i) that is accessible from state i, for each i = 1, 2, . . . , 7? (d) Identify which states are transient and which states are recurrent. For each recurrent class, state whether it is periodic (and give the period) or aperiodic. (e) What is the minimum number of transitions with nonzero probability that must be added so that all seven states form a single recurrent class?