(a) Explanation Given: A humanities major would shift to a science major with a probability of 20% and would be undecided with a probability of 45%. A science major changes to being a humanities major with a probability 0.15 and becomes undecided with a probability of 0.35. An undecided student would become a science major with a probability 0.3 and a humanities major with a probability 0.5. Explanation: The probabilites for the student shifting from one major to another can be represented with the help of a transition matrix. Three majors would correspond to a square transition matrix of order 3. Define the following variables: S : Science major H : Humantites major U : Undecided Use the provided information to set up the transition matrix as P . S H U P = S H U [ a 11 0.15 0.35 0.2 a 22 0.45 0.3 0.5 a 33 ] As the sum of all the entries of the transition matrix needs to be 1, the unknown diagonal elements can be calcuated as a 11 + 0.15 + 0.35 = 1 a 11 + 0.5 = 1 a 11 = 1 − 0.5 = 0.5 0.2 + a 22 + 0.45 = 1 a 22 = 1 − 0.25 − 0.45 = 1 − 0.65 = 0.35 0.3 + 0.5 + a 33 = 1 a 33 = 1 − 0.3 − 0.5 = 1 − 0.8 = 0.2 This results in the following transition matrix S H U P = S H U [ 0.5 0.15 0.35 0.2 0.35 0.45 0.3 0.5 0.2 ] The long-range probabilities for the student to be in each major 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 ] [ 0.5 0.15 0.35 0.2 0.35 0.45 0.3 0.5 0.2 ] = [ v 1 v 2 v 3 ] Simplify the left hand side to obtain [ 0.5 v 1 + 0.2 v 2 + 0.3 v 3 0.15 v 1 + 0.35 v 2 + 0.5 v 3 0.35 v 1 + 0.45 v 2 + 0.2 v 3 ] = [ v 1 v 2 v 3 ] This gives ride to the following system of equations 0.5 v 1 + 0.2 v 2 + 0.3 v 3 = v 1 0.15 v 1 + 0.35 v 2 + 0.5 v 3 = v 2 0.35 v 1 + 0.45 v 2 + 0.2 v 3 = v 3 This can be simplified to − 0.5 v 1 + 0.2 v 2 + 0.3 v 3 = 0 0.15 v 1 − 0.65 v 2 + 0.5 v 3 = 0 0.35 v 1 + 0.45 v 2 − 0.8 v 3 = 0 Also, use the property that v 1 , v 2 , and v 3 are entries of a probability vector so they should sum to 1 (b) To determine The comparison of the result obtained with the result of exercise 23.

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 10.2, Problem 37E

(a)

To determine

The long-range probability of the fractions of students in each of the three majors.

(b)

To determine

The comparison of the result obtained with the result of exercise 23.

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Chapter 10.2, Problem 36E

Chapter 10.2, Problem 38E

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The long-range probability of the fractions of students in each of the three majors.

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The long-range probability of the fractions of students in each of the three majors.

The comparison of the result obtained with the result of exercise 23.

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Chapter 10 Solutions