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).
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).
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 12EQ:
12. Robots have been programmed to traverse the maze shown in Figure 3.28 and at each junction...
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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
(c) Find the conditional probability P(X1 = 1|X2 = 1).
(d) Find limn→∞ P(Xn = 1).
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