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Math Help - Discrete time Markov Chain - Long-term frequency

  1. #1
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    Discrete time Markov Chain - Long-term frequency




    I just want to confirm something regarding this question. Since self transitions are allowed does that mean, for example, state 1 has 1/2 probability of self transitioning and 1/2 probability of transitioning to state 4? Similarly for state 4, it has 1/4 probability of transitioning to itself, state 1, state 5, and state 7, respectively, correct?
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  2. #2
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    Re: Discrete time Markov Chain - Long-term frequency

    Hello, usagi_killer!


    Code:
          *-------*-------*-------*
          |       |       |       |
          |   1   |   2   |   3   |
          |       |       |       |
          *--   --*--   --*--   --*
          |       |       |       |
          |   4       5       6   |
          |       |       |       |
          *--   --*--   --*--   --*
          |       |       |       |
          |   7   |   8   |   9   |
          |       |       |       |
          *-------*-------*-------*
    I just want to confirm something regarding this question.
    Since self-transitions are allowed does that mean, for example,
    state 1 has 1/2 probability of self transitioning and 1/2 probability of transitioning to state 4?

    Similarly for state 4, it has 1/4 probability of transitioning to itself,
    state 1, state 5, and state 7, respectively, correct? .Yes!

    I assume that we are to find the transition matrix.


    . . . \begin{bmatrix}\frac{1}{2} &0&0& \frac{1}{2}&0&0&0&0&0\\ 0& \frac{1}{2} &0&0& \frac{1}{2} &0&0&0&0 \\ 0&0&\frac{1}{2}&0&0&\frac{1}{2} &0&0&0 \\ \frac{1}{4} &0&0& \frac{1}{4} & \frac{1}{4} &0& \frac{1}{4} &0&0 \\ 0&\frac{1}{5} &0&\frac{1}{5}&\frac{1}{5}&\frac{1}{5} &0&\frac{1}{5} & 0 \\ 0&0&\frac{1}{4} &0& \frac{1}{4}&\frac{1}{4} &0&0&\frac{1}{4} \\ 0&0&0&\frac{1}{2} &0&0&0&\frac{1}{2}&0\\ 0&0&0&0&\frac{1}{2}&0&0&\frac{1}{2}&0 \\ 0&0&0&0&0&\frac{1}{2}&0&0&\frac{1}{2}  \end{bmatrix}

    Do we agree?
    Thanks from usagi_killer
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  3. #3
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    Re: Discrete time Markov Chain - Long-term frequency

    it's not stated that way but it's a legitimate assumption and in the absence of a further specification of the probabilities involved it's the one you should go with.

    But in general "at random" does not equate to "with equal probability"
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