Explanation Given: The transition probabilities from one state to another is, E l e c . G a s O i l S o l a r E l e c t r i c i t y .70 0 .10 0 N a t u r a l g a s .15 .90 .10 .05 F u e l o i l .05 .02 .75 .05 S o l a r e n e r g y .10 .08 .05 .09 Approach: Let T be the transition matrix. Then the steady-state distribution vector is given by solving the vector equation T X = X together with the condition that the sum of the elements of the vector X be equal to 1 . Calculation: Let X = [ w x y z ] be the steady-state distribution vector. Then, X satisfies the equation as mentioned in the approach part Here T is the given transition probability matrix; that is, T = [ 0.70 0 0.10 0 0.15 0.90 0.10 0.05 0.05 0.02 0.75 0.05 0.10 0.08 0.05 0.90 ] Solve the equation T X = X as follows, [ 0.70 0 0.10 0 0.15 0.90 0.10 0.05 0.05 0.02 0.75 0.05 0.10 0.08 0.05 0.90 ] [ w x y z ] = [ w x y z ] [ 0.7 w + 0.1 y 0.15 w + 0.90 x + 0.10 y + 0.05 z 0.05 w + 0.02 x + 0.75 y + 0.05 z 0.10 w + 0.08 x + 0.05 y + 0.90 z ] = [ w x y z ] The equations are given as, 0.7 w + 0.1 y = w 0.15 w + 0.90 x + 0.10 y + 0.05 z = x 0.05 w + 0.02 x + 0.75 y + 0.05 z = y 0.10 w + 0.08 x + 0.05 y + 0.90 z = z } ... ( 1 ) The system of equations ( 1 ) is simplified as, 30 w − 10 y = 0 … ( 2 ) 15 w − 10 x + 10 z + 5 z = 0 … ( 3 ) 5 w + 2 x − 25 y + 5 z = 0 … ( 4 ) 10 w + 8 x + 5 y − 10 z = 0 … ( 5 ) Also the sum of the elements is equal to one; that is, w + x + y + z = 1 … ( 6 ) Multiply equation ( 6 ) with 10 and add equation ( 5 ) to it as, 10 w + 10 x + 10 y + 10 z = 10 10 w + 8 x + 5 y − 10 z = 0 _ 20 w + 18 x + 15 y = 10 20 w + 18 x

Finite Mathematics for the Managerial, Life, and Social Sciences-Custom Edition

11th Edition

ISBN: 9781305283831

Author: Tan

Publisher: Cengage Learning

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Chapter 9.2, Problem 22E

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The percentage of homeowners who will use solar energy as their major source of heating fuel.

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Chapter 9.2, Problem 21E

Chapter 9.2, Problem 23E

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Chapter 9 Solutions

Finite Mathematics for the Managerial, Life, and Social Sciences-Custom 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 percentage of homeowners who will use solar energy as their major source of heating fuel.

Solution Summary: The author explains that the percentage of homeowners who will use solar energy as their major source of heating fuel is 42.8mathrm%.

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To find: The percentage of homeowners who will use solar energy as their major source of heating fuel.

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Chapter 9 Solutions

To find:

The percentage of homeowners who will use solar energy as their major source of heating fuel.