(a) Explanation Given: Consider the statement “At a particular two-year college, a student has a probability of 0.25 of flunking out during a given year, a 0.15 probability of having to repeat the year, and a 0.6 probability of finishing the year.” Use the states below: 1: Freshman 2: Sophomore 3: Has flunkded out 4: Has graduated Explanation: Using the above statements, the transition matrix will be formed as: 1 2 3 4 1 2 3 4 [ 0.15 0.6 0.25 0 0 0.15 0.25 0.6 0 0 1 0 0 0 0 1 ] We can see that the element P 3 , 3 = 1 and P 4 , 4 = 1 . Thus, it means that states 3 and 4 are absorbing state of the matrix. So, rewite the matrix in the form of: [ I m O R Q ] where I m is the identity matrix with m as the absorbing state and O is the matrix of all zeroes. So, the matrix can be written as: 1 2 3 4 1 2 3 4 [ 1 0 0 0 0 1 0 0 0.25 0 0.15 0.6 0.25 0.6 0 0.15 ] From comparing the above equation: R = [ 0.25 0 0.25 0.6 ] and, Q = [ 0.15 0.6 0 0.15 ] The fundamental matrix is: F = [ I 2 − Q ] − 1 As Q is of order 2, so identity matrix is order 2. So, I = [ 1 0 0 1 ] Therefore, F = ( [ 1 0 0 1 ] − [ 0.15 0.6 0 0.15 ] ) − 1 = [ 0 (b) To determine The probability that the freshman graduated. (c) To determine The expected number of years.

11th Edition

ISBN: 9780321979438

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

Publisher: PEARSON

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Chapter 10.3, Problem 28E

(a)

To determine

Find F and FR.

(b)

To determine

The probability that the freshman graduated.

(c)

To determine

The expected number of years.

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Chapter 10.3, Problem 27E

Chapter 10.3, Problem 29E

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Chapter 10 Solutions

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Find F and F R .

Solution Summary: The author explains how the transition matrix will be formed based on the following statements: Freshman, Sophomore, and Graduate.

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Find F and FR.

The probability that the freshman graduated.

The expected number of years.

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Chapter 10 Solutions