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Find all Invariant Probability Measures for P (Markov Chain)

  1. Mar 23, 2012 #1
    1. The problem statement, all variables and given/known data
    Find all Invariant Probability Measures for P (Markov Chain)
    E = {1,2,3,4,5}

    The screenshot below has P and my attempted solution. I am wondering if it acceptable to have infinitely many answers ("all" seems to indicate that is acceptable). Basically, I had too many unknowns and too few equations. Does my work look right?
    http://i.minus.com/ioEfrJKWamvpR.JPG [Broken]

    edit: I should add also that pi_1 >= 0 and pi_1 <= 1/2 to force the probabilities to be between 0 and 1. And after a bit of thought, I feel more comfortable with my answer. It seems to embody the fact that if I start at state 3, I stay there forever. Hence, pi_1 = 0 => steady-state probabilities are [0,0,1,0,0]. I can also be stuck in the recurrent class {1,5} in which case pi_1 = 1/2, and I never reach 3. I.e. pi = [1/2,0,0,0,1/2]. And there are many possibilities in between if I start in state 4 etc. where I have a finite probability of being sucked into the recurrent class or into the absorbing class. So pi = [nonzero, 0, nonzero, 0, nonzero] which the equation can handle.

    edit2: Upon further thought, though, it seems there should only be 3 invariant probability measures. One if I start in state 3, one if I start in state one or five, one if I start in state 2 or 4 (since 2 a.s. goes to 4).

    I believe the answers are:
    start in 1 or 5: [1/2 0 0 0 1/2] (so pi_1 = 1/2)
    start in 3: [0 0 1 0 0] (so pi_1 = 0)
    start in 2 or 4: [1/4 0 1/2 0 1/4] (so pi_1 = 1/4)
     
    Last edited by a moderator: May 5, 2017
  2. jcsd
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