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Math Help - Prove Bernoulli Process is Markov Chain

  1. #1
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    Prove Bernoulli Process is Markov Chain

    Let Z_1, Z_2, ... be a Bernoulli Process with success parameter p. Prove if

    Y_0, Y_1,..., where Y_{k+1} = max(0, Y_k + 2Z_{k+1} - 1) for k = 0,1,... and Y_0 = 0

    and

    X_0, X_1,.. where X_{k+1} = Z_{k+1} + Z_k for k = 0,1,..., and X_0 = 0

    are Markov chains.
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  2. #2
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    Does anyone have any idea how to go about these?
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  3. #3
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    X_2 = Z_2 + Z_1

    Let us suppose X_2 = 1 then, either Z_2=1 \mbox{ and } Z_1=0 \mbox{or} Z_1=0 \mbox{ and } Z_2=1 with 0.5 probability of each case.

    Given this, and knowing that X_3 = Z_3 + Z_2

    P(X_3 = 1|X_2 = 1) = p*0.5 + (1-p)*0.5 = 0.5

    However, if we know that X_1 = Z_1 + Z_0 = 2 we know that Z_1 = 1

    Taking this into account, we get that, X_1 = 2 and X_2 = 1, imply Z_2 = 0.

    Therefore,

    P(X_3 = 1 | X_2 = 1, X_1 = 2) = p

    It doesn't seem to be a Markov Chain
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