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Thread: Transition Matrix - Stationary Distribution

  1. #1
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    Transition Matrix - Stationary Distribution

    Consider the one-step transition matrix

    $\displaystyle P = \begin{pmatrix} q & p & 0 \\0 & q & p \\ p & 0 & q \\ \end{pmatrix} $

    where $\displaystyle q = 1 - p$ and $\displaystyle 0 < p < 1$.

    Find all stationary distributions of a chain with state space $\displaystyle E = \{0,1,2\}$ and one-step transition matrix P.

    Are any stationary distributions also limiting distributions?

    Thanks
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  2. #2
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    My mid-term is in two days and I really need to understand how to calculate this sort of question.

    I have.....
    The equation $\displaystyle \pi = \pi P$ becomes

    $\displaystyle \pi_0 = q \pi_0 + p \pi_1 $

    $\displaystyle \pi_1 = q \pi_1 + p \pi_2 $

    $\displaystyle \pi_2 = p \pi_0 + q \pi_2$

    I really dont know how to get the solutions from here.
    Last edited by RoyalFlush; Sep 14th 2009 at 06:41 AM.
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  3. #3
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    i think,

    $\displaystyle \pi_0 + \pi_1 + \pi_2 = 1 $

    so you can just make one of the $\displaystyle \pi $ the subject and then sub it into the 3 equations that you found yeah and continue until you found all the values of $\displaystyle \pi $
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  4. #4
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    I think my talent for mathematics has just about run its course

    I get

    $\displaystyle \pi_2 = \frac{p-p\pi_1}{1-p+q}$, $\displaystyle \pi_1 = \frac{\pi_2 - p - q\pi_2}{1-q-p}$ and $\displaystyle \pi_0 = \frac{\pi_1 - q\pi_1 - \pi_2 +p+q\pi_2}{1-q}$

    Im confused with the difference between the limiting distribution and the stationary distribution???? Can anyone please explain this?
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