Explanation Given: T = [ .3 .1 .1 .3 .5 .2 .4 .4 .7 ] . Approach: The probability distribution, in the long run, is given by the steady-state distribution vector X , which can be found with the help of the following equation: T X = X . Calculation: Let, steady-state distribution vector be X = [ x y z ] , T X = X [ .3 .1 .1 .3 .5 .2 .4 .4 .7 ] [ x y z ] = [ x y z ] Obtain the linear system { .3 x + .1 y + .1 z = x .3 x + .5 y + .2 z = y .4 x + .4 y + .7 z = z The equation of this linear system is equivalent to { 7 x − y − z = 0 3 x − 5 y + 2 z = 0 4 x + 4 y − 3 z = 0 . Also, x + y + z = 1 Solving the linear system by Guass-jordan method. The augmented matrix of the equation is [ 1 1 1 1 7 − 1 − 1 0 3 − 5 2 0 4 4 − 3 0 ] The augmented matrix is transformed into its reduced row-echelon form. Perform the transformation R 2 → R 2 − 7 R 1 . [ 1 1 1 1 7 − 7 ( 1 ) − 1 − 7 ( 1 ) − 1 − 7 ( 1 ) 0 − 7 ( 1 ) 3 − 5 2 0 4 4 − 3 0 ] → [ 1 1 1 1 0 − 8 − 8 − 7 3 − 5 2 0 4 4 − 3 0 ] Perform the transformation R 3 → R 3 − 3 R 1 . [ 1 1 1 1 0 − 8 − 8 − 7 3 − 3 ( 1 ) − 5 − 3 ( 1 ) 2 − 3 ( 1 ) 0 − 3 ( 1 ) 4 4 − 3 0 ] → [ 1 1 1 1 0 − 8 − 8 − 7 0 − 8 − 1 − 3 4 4 − 3 0 ] Perform the transformation R 4 → R 4 − 4 R 1 . [ 1 1 1 1 0 − 8 − 8 − 7 0 − 8 − 1 − 3 4 − 4 ( 1 ) 4 − 4 ( 1 ) − 3 − 4 ( 1 ) 0 − 4 ( 1 ) ] → [ 1 1 1 1 0 − 8 − 8 − 7 0 − 8 − 1 − 3 0 0 − 7 − 4 ] Perform the transformation R 3 → R 3 − R 2 . [ 1 1 1 1 0 − 8 − 8 − 7 0 − 8 − ( − 8 ) − 1 − ( − 8 ) − 3 − ( − 7 ) 0 0 − 7 − 4 ] → [ 1 1 1 1 0 − 8 − 8 − 7 0 0 7 4 0 0 − 7 − 4 ] Perform the transformation R 2 → − 1 8 R 2

Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition

11th Edition

ISBN: 9781285464657

Author: Soo T. Tan

Publisher: Brooks Cole

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Chapter 9 Solutions

Finite Mathematics for the Managerial, Life, and Social Sciences, 11th Edition

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 distribution of cars as per their sizes in the long-run.

Solution Summary: The author explains how the probability distribution is given by the steady-state distribution vector X, which can be found with the help of the following equation.

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To find: The distribution of cars as per their sizes in the long-run.

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Chapter 9 Solutions

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The distribution of cars as per their sizes in the long-run.