Consider a random walk {Xn : n = 0, 1, . . . } on the integer line S = {. . . , −2, −1, 0, 1, 2, . . . } and suppose that Pi,i+1 =1/3 and Pi,i-1 = 2/3 for i ∈ S. a) Draw a transition state diagram. b)True or False? All states are recurrent. Justify your answer. c) What is the long-run proportion of time spent in each state, πi? d) If Pi,i+1 = Pi,i-1 = 1/2 instead, how do parts (b) and (c) change? please answer parts a, b, c and d.

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