answer all questions with explanation

4. Simple Random Walk on a Circle. Let N 2 2 be an integer. We can consider {0, 1, - -- , N l} to be a \"circle\" by assuming that N 1 is adjacent to 0 as well N 2. Let Xn be simple random walk on the circle. The transition probabilities are Pas-1 = MP1,}; = 0.5,}6 =1,--- ,N laPO,N1 = 'pN_1,0 = 0.5. Let N = 6, (a) (5 points) What is the transition matrix P? (b) (5 points) Is there a limiting probability vector? If yes, what is it? If no, what is the period? (c) (5 points) Is there any invariant probability distribution? If yes, what is it? Is it unique? If no, Why? (d) (5 points) Is there any recurrent state? If yes, what is it? If 110, why? (e) (5 points) Is there any transient state? If yes, what is it? If no, why