Explanation Explanation Theorem used: Theorem 5: Let P be the transition matrix for an irreducible Markov chain with period d > 1 and q be the steady-state vector for Markov chain. Then, for any initial probability vector X 0 , lim n → ∞ 1 d ( P n + 1 + P n + 2 + ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ P n + d ) X 0 = q . Calculation: First calculate the transition matrix form the given transition diagram. Observe that there are 5 states and each state has some direction to visits other state. Start from state 1, it visits state 3 with probability 1. Similarly, the state 2 visits the state 1 with probability 1. And the states 3 visits state 4 and state 2 with probability 1 2 . Moreover, the state 4 visits 5 and the state 5 visits 3 each with probability 1. Based upon above computation, obtain the transition matrix for the simple random walk as follows. P = [ 0 1 0 0 0 0 0 1 2 0 0 1 0 0 0 1 0 0 1 2 0 0 0 0 0 1 0 ] . From the transition matric the irreducible Markov chain is { 1 , 2 , 3 , 4 , 5 } . Since, the any state returns back to its original point in 3 or more step after visiting the rest of the states. Thus, the period of the Markov chain { 1 , 2 , 3 , 4 , 5 } is d = 3 (which is d > 1 ). Use the definition state above and obtain the steady-state vector as follows. The given transition matrix is P = [ 0 1 0 0 0 0 0 1 2 0 0 1 0 0 0 1 0 0 1 2 0 0 0 0 0 1 0 ] . From the definition stated above, it is observed that the steady-state vector q is given by solving 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 system ( P − I ) q = 0 to obtain the steady-state vector q . The system ( P − I ) can be written as follows, [ 0 1 0 0 0 0 0 1 2 0 0 1 0 0 0 1 0 0 1 2 0 0 0 0 0 1 0 ] − [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ] = [ − 1 1 0 0 0 0 − 1 1 2 0 0 1 0 − 1 0 1 0 0 1 2 − 1 0 0 0 0 1 − 1 ] Thus, the augmented matrix is [ − 1 1 0 0 0 : 0 0 − 1 1 2 0 0 : 0 1 0 − 1 0 1 : 0 0 0 1 2 − 1 0 : 0 0 0 0 1 − 1 : 0 ] . To obtain q , the row-echelon form of the augmented matrix is obtained as follows. [ − 1 1 0 0 0 : 0 0 − 1 1 2 0 0 : 0 1 0 − 1 0 1 : 0 0 0 1 2 − 1 0 : 0 0 0 0 1 − 1 : 0 ] ~ [ 1 0 0 0 − 1 : 0 0 1 0 0 − 1 : 0 0 0 1 0 − 2 : 0 0 0 0 1 − 1 : 0 0 0 0 0 − 1 : 0 ] So the general solution of the steady-state vector is, q = q 5 [ 1 1 2 1 1 ] , q 5 is an arbitrary variable with any nonzero value. Since, q be a probability vector, hence the state vectors must sum to 1. The sum of the entries in the vector is 6. Divide the entries of q by 6 with q 5 = 1 and obtain the steady-state vector as follows, q = [ 1 6 1 6 1 3 1 6 1 6 ] Thus, the steady-state vector is q = [ 1 6 1 6 1 3 1 6 1 6 ] . For verification of the Theorem 5, here need to calculate the higher power of the transition matrix P . Calculation of the higher power of the transition matrix as follows. P 2 = P × P = [ 0 1 0 0 0 0 0 1 2 0 0 1 0 0 0 1 0 0 1 2 0 0 0 0 0 1 0 ] × [ 0 1 0 0 0 0 0 1 2 0 0 1 0 0 0 1 0 0 1 2 0 0 0 0 0 1 0 ] = [ 0 0 1 2 0 0 1 2 0 0 0 1 2 0 1 0 1 0 1 2 0 0 0 1 2 0 0 1 2 0 0 ] Thus, P 2 = [ 0 0 1 2 0 0 1 2 0 0 0 1 2 0 1 0 1 0 1 2 0 0 0 1 2 0 0 1 2 0 0 ] . Similarly multiplied P 2 with P and obtain P 3 as follows. P 3 = P 2 × P = [ 0 0 1 2 0 0 1 2 0 0 0 1 2 0 1 0 1 0 1 2 0 0 0 1 2 0 0 1 2 0 0 ] × [ 0 1 0 0 0 0 0 1 2 0 0 1 0 0 0 1 0 0 1 2 0 0 0 0 0 1 0 ] = [ 1 2 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 2 ] Hence, P 3 = [ 1 2 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 0 0 0 1 2 0 1 2 0 1 2 0 0 0 1 2 ]

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

See similar textbooks

Videos

Question

Chapter 10.4, Problem 24E

To determine

To verify: The statement of Theorem 5 for the Markov chain in Exercise 8 by taking powers of the transition matrix.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 10.4, Problem 23E

Chapter 10.4, Problem 25E

Students have asked these similar questions

Consider a Markov random process whose state transition diagram is shown in figure below. Write the state transition matrix for the Markov process whose corresponding state transition diagram is shown in the above figure. List the pairs of communicating states. List 2 pairs of accessible states and 2 pairs of inaccessible states. List all the transient states List all the recurrent states Identify the classes of the Markov chain and list the closed and non-closed Find P[X2 = 2 | X1 = 3] Find P[X6 = 4, X5 = 3, X4 = 2, X3 =3, X2 = 2, X1 = 1, X0 =1]. where Xt denotes the state of the random process at time instant t. The initial probability distribution is given by X0 = [2/3 0 0 0 0 0 1/3]. KIndly request you to refer to the screenshot for the figure

Please find the transition matrix for this Markov process

Chapter 10 Solutions

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...

Knowledge Booster

$Algebra$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, algebra and related others by exploring similar questions and additional content below.

To verify: The statement of Theorem 5 for the Markov chain in Exercise 8 by taking powers of the transition matrix.

Solution Summary: The author explains the statement of Theorem 5 for the Markov chain by taking powers of the transition matrix.

Videos

To verify: The statement of Theorem 5 for the Markov chain in Exercise 8 by taking powers of the transition matrix.

Want to see the full answer?

Chapter 10 Solutions