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Math Help - Transition Matrix question

  1. #1
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    Transition Matrix question

    a housekeeper buys three kinds of cereal; A,B,C. She never buys the same cereal in successive weeks. if she buys cereal A, then the next week she buys cereal B. However, if she buys either B or C, then the next week she is three times as likely to buy A as the other brand.

    a) find the transition matrix
    b) in the long run, how often does she buy each of the three brands?


    for a. i got
    T=
    [0 1 0]
    |.6 0 .2|
    [.6 .2 0]

    im not sure how to do the second part or if i got the right transition matrix for part a. help?
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  2. #2
    Ond
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    I think the matrix should be:

    \left( \begin{array}{ccc}<br />
0 & 1 & 0 \\<br />
& \\<br />
0,75 & 0 & 0,25 \\<br />
& \\<br />
0,75 & 0,25 & 0 \end{array}\right)

    Isn't then an initial-state matrix needed to calculate the rest?

    Anyway, if you have an initial state matrix, called S_0 and then the transition matrix above, called P, the first state matrix, S_1 would be calculated as follows: S_1=S_0P. The second state matrix as: S_2=S_0P^2. The k-th state matrix: S_k=S_0P^k

    I'm not quite sure about those Markov chains, but I hope that helps.
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  3. #3
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    Quote Originally Posted by jellosalade View Post
    a housekeeper buys three kinds of cereal; A,B,C. She never buys the same cereal in successive weeks. if she buys cereal A, then the next week she buys cereal B. However, if she buys either B or C, then the next week she is three times as likely to buy A as the other brand.

    a) find the transition matrix
    b) in the long run, how often does she buy each of the three brands?


    for a. i got
    T=
    [0 1 0]
    |.6 0 .2|
    [.6 .2 0]

    im not sure how to do the second part or if i got the right transition matrix for part a. help?
    Asked here: http://www.mathhelpforum.com/math-he...on-matrix.html

    Thread closed.
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