A DTMC (Xn)nezt with state space $ = {1,2,3,4} has the following transition matrix a P= 1/2 0 d 2/3 0 1/2 0 0 0 b 0 0 с

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Please do part (j) - part(m) specific due to guidelines
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A DTMC (Xn)nezt with state space $ = {1,2,3,4} has the following transition matrix a P= 1/2 0 d 2/3 0 1/2 0 0 0 b 0 0 с 1/2 0 (a) Find the values of a, b,c,d. (b) Draw the transition diagram for this chain. (c) Find P(X5 = 4X4 = 3). (d) Find P(X5 = 4 X2 = 1, X4 = 3). (e) Find P(X3 = 4 X 3 = 1). (f) Find P(X4 = 3|Xo = 1). (g) Does the chain have any absorbing state(s)? (h) Is the chain irreducible? (i) Is the chain periodic? If so, what is the period? 6) Is the chain transient, null-recurrent, or positive recurrent? (k) Is the process reversible? Why, or why not? (1) Find the long run proportion of time spent in state 4. (m) Find the expected time to reach state 2, starting from state 1. (n) Find the expected time to reach state 3, starting from state 1. (0) If the initial distribution is (3/8, 3/8, 1/8, 1/8), does the limiting distribution exist? If so, find it. (p) Give a simple and accurate estimate for the matrix P1000, (9) Starting from state 2, find the probability of hitting state 4 before state 1.