Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · ·...

Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · } and transition matrix (pij ) given by pij = 1 2 if j = i − 1 1 2 if j = i + 1, i ≥ 1, and p00 = p01 = 1 2 . Find P{X0 ≤ X1 ≤ · · · ≤ Xn|X0 = i}, i ≥ 0

. Q2. Consider the Markov chain given in Q1. Find P{X1, X2, X3 ∈/ A, X4 = 4|X0 = 2}, A = {0, 1}.

Q3. Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · } and transition matrix (pij ). Show that P{X0 ≤ X1 ≤ · · · ≤ Xn|X0 = i} = X i≤i1≤i2···≤in−1≤in pii1 pi1i2 · · · pin−1in .

Q4. Consider a Markov chain {Xn|n ≥ 0} with state space S = {0, 1, · · · and transition matrix (pij ). Show that for i, j ∈ Ac , P{X1, · · · , Xn−1 ∈/ A, Xn = j|X0 = i} = q n ij (A), where Qn A = (q n ij (A)), QA = (qij (A))i,j∈Ac , qij (A) = pij if i, j ∈ Ac

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orchestra answered 1 year ago

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