Explanation Given: The transition matrix: [ Dogkins Long Dog Dogkins 0.8 0.2 Long Dog 0.45 0.55 ] Explanation: It is provided that the initial market share of Dogkins is 35% and of Long Dog is 65%. So, the transition matrix is: P = [ 0.8 0.2 0.45 0.55 ] The above matrix represents the probability of hot dogs by the companies after one campaign. The long-term distribution would correspond to the equilibrium vector . A vector v is said to be an equilibrium vector for a transition matrix P , if the following condition holds: v P = v Let the vector v be defined as follows, so that the multiplication holds. [ v 1 v 2 ] Then, v P = v [ v 1 v 2 ] [ 0.8 0.2 0.45 0.55 ] = [ v 1 v 2 ] Simplify the left-hand side to obtain: [ 0.8 v 1 + 0.45 v 2 0.2 v 1 + 0.55 v 2 ] = [ v 1 v 2 ] This gives rise to the following system of equations: 0.8 v 1 + 0.45 v 2 = v 1 0.2 v 1 + 0.55 v 2 = v 2 This can be simplified to: − 0.2 v 1 + 0.45 v 2 = 0 0

Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

11th Edition

ISBN: 9780133886818

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

Publisher: PEARSON

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Chapter 10, Problem 44RE

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Chapter 10, Problem 43RE

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Finite Mathematics, Books a la Carte Plus MyLab Math Access Card Package (11th Edition)

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Solution Summary: The author explains that the prediction of the long-range market share for Dogkins is 69.23%.

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