Explanation Approach: If T is a regular stochastic matrix, then the Steady state distribution vector X can be found by solving the vector equation, T X = X together with the condition that the sum of the elements of the vector X should be equal to 1. Calculation: Consider the matrix, T = [ 0.3 0.2 0.1 0.3 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.3 0.2 0.2 0.1 0.3 0.2 0.3 0.2 0.3 0.2 0.2 0.1 0.3 ] Let, X = [ x y z u v ] Be a steady state vector associated with the Markov process, the condition T X = X translates into the matrix equation, [ 0.3 0.2 0.1 0.3 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.3 0.2 0.2 0.1 0.3 0.2 0.3 0.2 0.3 0.2 0.2 0.1 0.3 ] [ x y z u v ] = [ x y z u v ] Solve the above matrix equation, [ 0.3 x + 0.2 y + 0.1 z + 0.3 u + 0.1 v 0.2 x + 0.1 y + 0.2 z + 0.1 u + 0.2 v 0.1 x + 0.2 y + 0.3 z + 0.2 u + 0.2 v 0.1 x + 0.3 y + 0.2 z + 0.3 u + 0.2 v 0.3 x + 0.2 y + 0.3 z + 0.1 u + 0.3 v ] = [ x y z u v ] Equivalently, the linear system of equation for above matrix equation is, 0.3 x + 0.2 y + 0.1 z + 0.3 u + 0.1 v = x 0.2 x + 0.1 y + 0.2 z + 0.1 u + 0.2 v = y 0.1 x + 0.2 y + 0.3 z + 0.2 u + 0.2 v = z 0.1 x + 0.3 y + 0.2 z + 0.3 u + 0.2 v = u 0.3 x + 0.2 y + 0.3 z + 0.1 u + 0.3 v = v The condition is that sum of X is 1 so, 0.7 x − 0.2 y − 0.1 z − 0.3 u − 0.1 v = 0 0

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Chapter 9.2, Problem 2TE

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The steady state vector of the matrix T=[0.30.20.10.30.10.20.10.20.10.20.10.20.30.20.20.10.30.20.30.20.30.20.20.10.3].

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Chapter 9.2, Problem 1TE

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Ch. 9.1 - What is a finite stochastic process? What can you...Ch. 9.1 - Prob. 2CQ Ch. 9.1 - Consider a transition matrix T for a Markov chain...Ch. 9.1 - Prob. 1E Ch. 9.1 - Prob. 2E Ch. 9.1 - Prob. 3E Ch. 9.1 - Prob. 4E Ch. 9.1 - Prob. 5E Ch. 9.1 - Prob. 6E Ch. 9.1 - Prob. 7E

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The steady state vector of the matrix T = [ 0.3 0.2 0.1 0.3 0.1 0.2 0.1 0.2 0.1 0.2 0.1 0.2 0.3 0.2 0.2 0.1 0.3 0.2 0.3 0.2 0.3 0.2 0.2 0.1 0.3 ] .

Solution Summary: The author explains that the steady state vector of the matrix T=left is X=X.

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The steady state vector of the matrix T=[0.30.20.10.30.10.20.10.20.10.20.10.20.30.20.20.10.30.20.30.20.30.20.20.10.3].

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Chapter 9 Solutions