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Math Help - Expected value of markov chain URGENT!

  1. #1
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    Expected value of markov chain URGENT!

    i have a 3 state (0,1,2) markov chain with transition matrix p=
    1/2, 1/3, 1/6
    0, 1/3, 2/3
    1/2, 0, 1/2
    and that p(x0= 0) = p(x0= 1)= 1/2

    and i need to fond E[X3]

    so i find p^3 by matrix multiplication to be:
    13/36, 11/54,47/108
    4/9, 4/27, 11/27
    5/12, 2/9, 13/36

    now the solution says the answer is 1/2(47/108)+1/2(11/54)+1/4(13/36)

    but i have no idea how this is obtained! please help!
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by mtlchris View Post
    i have a 3 state (0,1,2) markov chain with transition matrix p=
    1/2, 1/3, 1/6
    0, 1/3, 2/3
    1/2, 0, 1/2
    and that p(x0= 0) = p(x0= 1)= 1/2

    and i need to fond E[X3]

    so i find p^3 by matrix multiplication to be:
    13/36, 11/54,47/108
    4/9, 4/27, 11/27
    5/12, 2/9, 13/36

    now the solution says the answer is 1/2(47/108)+1/2(11/54)+1/4(13/36)

    but i have no idea how this is obtained! please help!
    The distribution over states after three steps where A is the single step transition matrix and x_0 the initial distribution over states is:

    x_3=x_0A^3=[1/2, 1/2, 0]A^3

    From this distribution you calculate the expectation for the state.

    CB
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