3. Suppose that (Xn)nzo is a Markov chain on a state space I = {1, 2,...

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3. Suppose that (Xn)nzo is a Markov chain on a state space I = {1, 2, 3, 4, 5}, initial distribution A = = (1/3, 0, 0, 1/6, 1/2) and stochastic matrix [1/4 1/4 1/4 0 1/4] 0 0 5/6 0 1/6 P = 1/3 0 1/5 1/6 0 1/6 1/2 1/4 1/4 1/4 1/4 1/5 1/5 1/5 1/5 a Calculate P(X₁ = 5) b Calculate P₁(X₁ = 5, X₂ = 1) c Calculate P₁(X10 = 5|X9 = 2) d Calculate P₁(X10 = 5|X9 = 2, X4 = 3) e Calculate P₁(X10 = 5, X₁1 = 1|X9 = 2, Xg = 3) f HARDER: Calculate P₁(X8 = 3, X10 = 5, X11 = 1|X9 = 2, X7 = 1) 3. Suppose that (Xn)nzo is a Markov chain on a state space I = {1, 2, 3, 4, 5}, initial distribution A = = (1/3, 0, 0, 1/6, 1/2) and stochastic matrix [1/4 1/4 1/4 0 1/4] 0 0 5/6 0 1/6 P = 1/3 0 1/5 1/6 0 1/6 1/2 1/4 1/4 1/4 1/4 1/5 1/5 1/5 1/5 a Calculate P(X₁ = 5) b Calculate P₁(X₁ = 5, X₂ = 1) c Calculate P₁(X10 = 5|X9 = 2) d Calculate P₁(X10 = 5|X9 = 2, X4 = 3) e Calculate P₁(X10 = 5, X₁1 = 1|X9 = 2, Xg = 3) f HARDER: Calculate P₁(X8 = 3, X10 = 5, X11 = 1|X9 = 2, X7 = 1)