8. (10 points) Let (X, )n=0,1,2.. be a Markov chain with transition graph 1 1/3 1/3 2 3 1/3 (a) Provide the transition probability matrix P. (b) Is the Markov chain irreducible? Is the Markov chain aperiodic? Give a reason for your answers. (c) Compute for all i, je S = (1, 2,3) the limit lim pa. n-00 i,j' Justify your answer! Does this limit depend on the initial starting value ? (d) Let 72 := min{n 2 1 : Xn = 2). Compute Ez[72]