4. Simple Markov Chain Let X0, X1, X2,... be a Markov chain with state space S = {1, 2, 3}, transition matrix 0.1 0.1 0.8 P = 0.2 0.4 0.4 0.3 0.7 0 and initial distribution aT = (0.1, 0.6, 0.3). Compute (a) P[X7 = 1|X6 = 3], (b) P[X9 = 3|X₁ = 1, X5 = 2, X7 = = 1], (c) P[X0 = 2|X₁ = 3], (d) E[X2].

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