Explanation Given Information: The stable distributions for the provided stochastic matrix [ .4 0 .9 0 1 0 .6 0 .1 ] are [ .6 0 .4 ] and [ 0.3 0.5 0.2 ] respectively. Formula used: The stable distribution for the matrix X = [ ] , could be determined by solving the system of linear equations as, 1. The sum of entries of X = 1 . 2. And A X = X . Proof: Consider the stable distributions, [ .6 0 .4 ] and [ 0.3 0.5 0.2 ] for the provided stochastic matrix, [ .4 0 .9 0 1 0 .6 0 .1 ] . Consider the stable distribution [ .6 0 .4 ] and [ 0.3 0.5 0.2 ] Here, the sum of entries of X is 1. 0.6 + 0 + 0.4 = 1 Similarly, 0.3 + 0.5 + 0.2 = 1 So, first condition is satisfied. From the formula for the stable distribution of the matrix X = [ ] , determined by solving the system of linear equations as, A X = X Thus, [ .4 0 .9 0 1 0 .6 0 .1 ] [ .6 0 .4 ] = [ .6 0 .4 ] and [ .4 0 .9 0 1 0 .6 0 .1 ] [ 0.3 0.5 0.2 ] = [ 0.3 0.5 0.2 ] So, these two matrices [ .6 0 .4 ] and [ 0.3 0.5 0.2 ] are stable distribution. When given arbitrary initial distribution for the value of 0 of the stochastic matrix, [ ] 0 ≠ [ 0.6 0 0.4 ] or [ 0.3 0.5 0.2 ] . Let take the initial distribution [ ] 0 = [ 0.5 0.3 0.2 ] . Now, calculate the matrix A 10 . First calculate the matrix A 2 ; A 2 = A ⋅ A = [ .4 0 .9 0 1 0 .6 0 .1 ] [ .4 0 .9 0 1 0 .6 0 .1 ] = [ 0.7 0 0.45 0 1 0 0.3 0 0.55 ] Calculate the matrix A 3 ; A 3 = A ⋅ A 2 = [

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Author: Larry Goldstein, David Schneider

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Chapter 8.2, Problem 25E

To determine

To prove: The matrix [0.600.4] and [0.30.50.2] are both stable distributions for the stochastic matrix

[.40.9010.60.1]. Also, explain the reason due to which this fact does not contradict the uniqueness of the stable distribution.

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Chapter 8.2, Problem 24E

Chapter 8.2, Problem 26E

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Chapter 8 Solutions

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Ch. 8.1 - 1. Is a stochastic matrix? Ch. 8.1 - 2. Learning Process An elementary learning process...Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - Prob. 4E Ch. 8.1 - In Exercises 1-6, determine whether or not the...Ch. 8.1 - Prob. 6E Ch. 8.1 - In Exercises 7–12, write a stochastic matrix...Ch. 8.1 - Prob. 8E

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To prove: The matrix [ 0.6 0 0.4 ] and [ 0.3 0.5 0.2 ] are both stable distributions for the stochastic matrix [ .4 0 .9 0 1 0 .6 0 .1 ] . Also, explain the reason due to which this fact does not contradict the uniqueness of the stable distribution.

Solution Summary: The author explains the stable distributions for the stochastic matrix left[cc.4& 0&

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To prove: The matrix [0.600.4] and [0.30.50.2] are both stable distributions for the stochastic matrix

[.40.9010.60.1]. Also, explain the reason due to which this fact does not contradict the uniqueness of the stable distribution.

Want to see the full answer?

Chapter 8 Solutions

To prove: The matrix [ 0.6 0 0.4 ] and [ 0.3 0.5 0.2 ] are both stable distributions for the stochastic matrix [ .4 0 .9 0 1 0 .6 0 .1 ] . Also, explain the reason due to which this fact does not contradict the uniqueness of the stable distribution.

Solution Summary: The author explains the stable distributions for the stochastic matrix left[cc.4& 0&

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To prove: The matrix [0.600.4] and [0.30.50.2] are both stable distributions for the stochastic matrix [.40.9010.60.1]. Also, explain the reason due to which this fact does not contradict the uniqueness of the stable distribution.

Want to see the full answer?

Chapter 8 Solutions

To prove: The matrix [0.600.4] and [0.30.50.2] are both stable distributions for the stochastic matrix

[.40.9010.60.1]. Also, explain the reason due to which this fact does not contradict the uniqueness of the stable distribution.