Science:Math Exam Resources/Courses/MATH152/April 2017/Question B 02
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Question B 02 

For each matrix below left, interpret the matrix as a probability transition matrix, and match it to its corresponding equilibrium probability on the right. One mark each. Note that answers in the list (I)(VII) may be a match more than once or not at all. (A) (B) (C) , (D) , (E) ,
(VI) not a probability transition matrix, (VII) equilibrium probability not in the list, (VIII) no equilibrium probability 
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Hint 

Let be the probability of moving to state if you are in state . Then the transition matrix matrix is defined by . When has a single eigenvector for the eigenvalue , we call the probability eigenvector as the equilibrium probability. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. First, we need to know that equilibrium probability is defined as a probability vector, i.e., the sum of its components is 1. Exclude from answer. Second, from the definition of the probability transition matrix, we have . (Total probability should be 1) So is not a transition probability matrix. (i.e., .) Recall that when has a single eigenvector for the eigenvalue , we call the probability eigenvector as the equilibrium probability. Since we only have transition matrices, the maximum number of eigenvectors corresponding to the eigenvalue is and in that case actually for any . i.e., . Therefore has no equilibrium probability. Since the remained probability transition matrices are not identity matrix and we only have limited number of options for equilibrium probabilities, instead of finding eigenvectors corresponding to the eigenvalue , we find equilibrium probabilities by trials and errors, : . . Thus corresponding equilibrium probability of is
. Thus corresponding equilibrium probability of is
Thus corresponding equilibrium probability of is
