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Math Help - Markov problem

  1. #1
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    Markov problem

    Hi everyone, any help in how to work out this problem would be much appreciated...

    If it is sunny today, there is a 80% chance it will be sunny tomorrow, and a 20% chance it will be rainy. If it is rainy today, there is a 40% chance it will be rainy tomorrow, and a 60% chance it will be sunny.

    In n days, how many of them will be sunny and how many will be rainy?



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  2. #2
    Newbie jaco's Avatar
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    This is an irreducible finite markov chain and
    P= 0.8 0.2
    0.6 0.4

    If you solve the limiting distribution you get that in the limit P(sunny)=0.75 and P(rain)=0.25

    therefore 0.75n days will be sunny and 0.25n days will be rainy.
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  3. #3
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    Thanks jaco, could you go through the working so I can get my head round it?
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  4. #4
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    you first have to get the limiting distributions which you can easily get by calculating P^n for n large enough and taking any row (all rows will be equal)
    or solving the matrix [(P-I)`|0] with last row substituted with 1's since probabilities must sum to 1.

    in this case:
    P-I=
    -0.2 0.2
    0.6 -0.6

    (P-I)`=
    -0.2 0.6
    0.2 -0.6

    solve for:
    -0.2 0.6 | 0
    1 1 | 1

    or -0.2x+0.6y=0 and x+y=1

    you will get x=0.75 and y=0.25 if you solve the 2 equations.
    these are your limiting probabilities.

    let me know if you need more help
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  5. #5
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    Thanks, that's extremely helpful
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  6. #6
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    One further question... How long, on average, would a sequence of sunny days be? And how long would a sequence of rainy days be?

    Is there a way of working this out?
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  7. #7
    Newbie jaco's Avatar
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    i suppose you can calculate that empirically using sas or similar program
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  8. #8
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    That's just a geometric distribution. The sequence of sunny days, is on average 5 days, and for rainy days, it's 5/3 days.
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