Explanation Given: The next day after a fair day would be fair with a probability of 0.6 and cloudy with a probability of 0.25. The next day after a rainy day would be cloudy with a probability of 0.4 and rainy with a probability of 0.25. The next day after a cloudy day would be cloudy with a probability of 0.35 and rainy with a probability of 0.25. Explanation: The probabilities for the weather of the next day can be represented by a transition matrix. Three classifications of weather would imply a 3 × 3 transition matrix. Define the following variables: F : fair weather C : cloudy weather R : rainy weather Use the provided information to set up the transition matrix P . F C R P = F C R [ 0.6 0.25 a 13 a 21 0.35 0.25 a 31 0.4 0.25 ] For a transition matrix, the sum of all entries for each row needs to be 1. Use this information to obtain the remaining values as follows: 0.6 + 0.25 + a 13 = 1 a 13 = 1 − 0.6 − 0.25 = 1 − 0.85 = 0.15 a 21 + 0.35 + 0.25 = 1 a 21 = 1 − 0.35 − 0.25 = 1 − 0.6 = 0.4 a 31 + 0.4 + 0.25 = 1 a 31 = 1 − 0.4 − 0.25 = 1 − 0.65 = 0.35 This gives the following transition matrix: F C R P = F C R [ 0.6 0.25 0.15 0.4 0.35 0.25 0.35 0.4 0.25 ] The long-range probability for the weather of the next day can be computed as the equilibrium vector for the transition matrix P . 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 v 3 ] Then, v P = v [ v 1 v 2 v 3 ] [ 0.6 0.25 0.15 0.4 0.35 0.25 0.35 0.4 0.25 ] = [ v 1 v 2 v 3 ] Simplify the left-hand side to obtain [ 0.6 v 1 + 0.4 v 2 + 0.35 v 3 0.25 v 1 + 0.35 v 2 + 0.4 v 3 0.15 v 1 + 0.25 v 2 + 0.25 v 3 ] = [ v 1 v 2 v 3 ] This leads to the following system of equations: 0.6 v 1 + 0.4 v 2 + 0.35 v 3 = v 1 0.25 v 1 + 0.35 v 2 + 0.4 v 3 = v 2 0.15 v 1 + 0.25 v 2 + 0.25 v 3 = v 3 This can be simplified to − 0.4 v 1 + 0.4 v 2 + 0.35 v 3 = 0 0.25 v 1 − 0.65 v 2 + 0.4 v 3 = 0 0.15 v 1 + 0.25 v 2 − 0.75 v 3 = 0 The system involving these four equations can be solved with the help of the Gauss–Jordan method. The augmented matrix for the system is [ − 0.4 0

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ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 10.2, Problem 40E

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The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

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Chapter 10.2, Problem 39E

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The proportion of days that are expected to be fair, cloudy, and rainy over the long term.

Solution Summary: The author explains how three classifications of weather would imply a 3times 3 transition matrix.

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