Explanation Given: The transition matrix [ ( 1 − q ) ( 2 − q − p ) ( 1 − p ) ( 2 − q − p ) ] ¯ . Explanation: A vector v is said to be an equilibrium vector for a transition matrix A , if the following condition holds: v A = v Let the vector v be defined as follows, so that the multiplication holds. [ v 1 v 2 ] Then v A = v [ v 1 v 2 ] [ p 1 − p 1 − q q ] = [ v 1 v 2 ] Simplify the left hand side to obtain [ p v 1 + ( 1 − q ) v 2 ( 1 − p ) v 1 + q v 2 ] = [ v 1 v 2 ] This gives ride to the following system of equations p v 1 + ( 1 − q ) v 2 = v 1 ( 1 − p ) v 1 + q v 2 = v 2 This can be simplified to ( p − 1 ) v 1 + ( 1 − q ) v 2 = 0 ( 1 − p ) v 1 + ( q − 1 ) v 2 = 0 This is basically just one equation ( 1 − p ) v 1 + ( q − 1 ) v 2 = 0 Simplify to obtain the following relation ( 1 − p ) v 1 = ( 1 − q ) v 2 ( 1 − p ) ( 1 − p ) v 1 = ( 1 − q ) ( 1 − p ) v 2 v 1 = ( 1 − q ) ( 1 − p ) v 2 Also, use the property that v 1 and v 2 are entries of a probability vector so they should sum to 1

11th Edition

ISBN: 9780321979438

Author: Margaret L. Lial, Raymond N. Greenwell, Nathan P. Ritchey

Publisher: PEARSON

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Chapter 10.2, Problem 21E

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The equilibrium vector corresponding to the transition matrix provided.

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Chapter 10.2, Problem 20E

Chapter 10.2, Problem 22E

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The equilibrium vector corresponding to the transition matrix provided.

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