Explanation Definition used: The steady-state vector : If P is a stochastic matrix, then a steady state vector (or equilibrium vector or invariant probability vector) for P is a probability vector q such that P q = q . Given: The number of vertices is 5 and edges are 6. Calculation: Every vertex is connected to other vertex through edges. Start from vertex 1, it can be walked towards three other vertices that are 2, 4 and 5, thus the probabilities for the walk from vertex 1 towards 2, 4 and 5 is 1 3 each. Similarly, the vertex 2 can be walked towards three other vertices that are 1, 3 and 5, thus the probabilities for the walk from vertex 2 towards 1, 3 and 5 is 1 3 each. The vertex 3 can be walked towards two other vertices that are 2 and 4, thus the probabilities for the walk from vertex 3 towards 2 and 4 is 1 2 each. The vertex 4 can be walked towards two other vertices that are 1 and 3, thus the probabilities for the walk from vertex 4 towards 1 and 3 is 1 2 each. The vertex 2 can be walked towards two other vertices that are 1 and 2, thus the probabilities for the walk from vertex 5 towards 1 and 2 is 1 2 each. Based upon above computation, obtain the transition matrix for the simple random walk as follows. P = [ 0 1 3 0 1 2 1 2 1 3 0 1 2 0 1 2 0 1 3 0 1 2 0 1 3 0 1 2 0 0 1 3 1 3 0 0 0 ] From the definition stated above, it is observed that, to obtain the steady-state vector q , solve the system P q = q where q = [ q 1 q 2 q 3 q 4 q 5 ] . It is clear that the system P q = q is equivalent to the homogeneous system ( P − I ) q = 0 . Hence, it is enough to solve the homogeneous system ( P − I ) q = 0 to obtain the steady-state vector q . The system ( P − I ) can be written as, [ 0 1 3 0 1 2 1 2 1 3 0 1 2 0 1 2 0 1 3 0 1 2 0 1 3 0 1 2 0 0 1 3 1 3 0 0 0 ] − [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] = [ − 1 1 3 0 1 2 1 2 1 3 − 1 1 2 0 1 2 0 1 3 − 1 1 2 0 1 3 0 1 2 − 1 0 1 3 1 3 0 0 − 1 ] Thus, the augmented matrix is [ − 1 1 3 0 1 2 1 2 : 0 1 3 − 1 1 2 0 1 2 : 0 0 1 3 − 1 1 2 0 : 0 1 3 0 1 2 − 1 0 : 0 1 3 1 3 0 0 − 1 : 0 ]

Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

5th Edition

ISBN: 9781323132098

Author: Thomas, Lay

Publisher: PEARSON C

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Chapter 10.2, Problem 13E

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To find: The fraction of time that the walk takes at each of the various states.

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Chapter 10.2, Problem 12E

Chapter 10.2, Problem 14E

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Thomas' Calculus and Linear Algebra and Its Applications Package for the Georgia Institute of Technology, 1/e

Ch. 10.1 - Fill in the missing entries in the stochastic...Ch. 10.1 - Prob. 2PP Ch. 10.1 - In Exercises 1 and 2, determine whether P is a...Ch. 10.1 - In Exercises 1 and 2, determine whether P is a...Ch. 10.1 - Prob. 3E Ch. 10.1 - Prob. 4E Ch. 10.1 - In Exercises 5 and 6, the transition matrix P for...Ch. 10.1 - Prob. 6E Ch. 10.1 - In Exercises 7 and 8, the transition matrix P for...Ch. 10.1 - In Exercises 7 and 8, the transition matrix P for...

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The fraction of time that the walk takes at each of the various states.

Solution Summary: The author explains the steady-state vector (or equilibrium vector or invariant probability vector) for P is q such that Pq=q.

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To find: The fraction of time that the walk takes at each of the various states.

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Chapter 10 Solutions