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Math Help - Solving a system of 4 equations

  1. #1
    Member mybrohshi5's Avatar
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    Solving a system of 4 equations

    I am trying to solve a problem for my stochastic process class and I am running into some problems solving the system of equations I have obtained from my transition matrix. It has been a while since I have taken algebra, or any class dealing with solving systems of equations, so please forgive me for my easy question

    I am trying to solve \pi * P = \pi where \pi = (\pi_0, \pi_1, \pi_2, \pi_3) and  P = \left( \begin{array}{cccc} 0.7&0.3&0&0 \\ 0&0&0.4&0.6 \\ 0.5&0.5&0&0 \\ 0&0&0.2&0.8  \end{array} \right)

    \pi * P = (.7\pi_0+.5\pi_2,.3\pi_0+.5\pi_2,.4\pi_1+.2\pi_3,.  6\pi_1+.8\pi_3) = (\pi_0, \pi_1, \pi_2, \pi_3)

    So I basically need help solving this system of equations:

     .7\pi_0+.5\pi_2= \pi_0
     .3\pi_0+.5\pi_2 = \pi_1
     .4\pi_1+.2\pi_3 = \pi_2
     .6\pi_1+.8\pi_3 = \pi_3

    Also note: \pi_0 + \pi_1 + \pi_2 + \pi_3 = 1

    Thank you in advance
    Last edited by mybrohshi5; March 6th 2012 at 05:21 PM.
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  2. #2
    MHF Contributor
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    Re: Solving a system of 4 equations

    Quote Originally Posted by mybrohshi5 View Post
     .7\pi_0+.5\pi_2= \pi_0
     .3\pi_0+.5\pi_2 = \pi_1
     .4\pi_1+.2\pi_3 = \pi_2
     .6\pi_1+.8\pi_3 = \pi_3
    "Unmess(!)" your equations by changing variables to a,b,c,d and multiplying by 10:
    7a + 5c = 10a
    3a + 5c = 10b
    4b + 2d = 10c
    6b + 8d = 10d

    Now play with that "silly" system
    Thanks from mybrohshi5
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  3. #3
    Member mybrohshi5's Avatar
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    Re: Solving a system of 4 equations

    Thank you Wilmer. That made it much easier to solve

    Answer should be a=1/4, b=c=3/20, and d=9/20

    To check: a+b+c+d = 1
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