Suppose that X0, X1, X2, ... form a Markov chain on the state space {1, 2}. Assume that P(X0 = 1) = P(X0 = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q11 = 1/2, Q12 = 1/2, Q21 = 1/3, Q22 = 2/3.(a) Find P(X2 = 1).(b) Find the conditional probability P(X2 = 1|X1 = 1).(c) Find the conditional probability P(X1 = 1|X2 = 1).(d) Find limn→∞ P(Xn = 1).

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Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.7: Applications

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Suppose that X₀, X₁, X₂, ... form a Markov chain on the state space {1, 2}. Assume that P(X₀ = 1) = P(X₀ = 2) = 1/2 and that the matrix of transition probabilities for the chain has the following entries: Q₁₁ = 1/2, Q₁₂ = 1/2, Q₂₁ = 1/3, Q₂₂ = 2/3.
(a) Find P(X2 = 1).
(b) Find the conditional probability P(X2 = 1|X1 = 1).
(c) Find the conditional probability P(X1 = 1|X2 = 1).
(d) Find lim_n→∞P(Xn = 1).

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