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Math Help - Stochastic process, expected absolute deviation

  1. #1
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    Stochastic process, expected absolute deviation

    X(t) is a Gaussian stochastic process with mean m_{X(t)}=0, ACF r_X(\tau)=\frac{1}{1+\tau^2}

    Determine the expected absolute deviation \epsilon = E(|X(t+1)-X(t)|)

    I have no idea where to begin except maybe setting Y(t)=X(t+1)-X(t), but then I have to figure out what the PDF f_{Y(t)}(y) is and I'm not sure how to proceed.
    Last edited by Mondreus; September 10th 2011 at 08:19 AM.
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  2. #2
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    Re: Stochastic process, expected absolute deviation

    The solution is that since X(t) is a Gaussian process, all linear combinations of X(t) are also Gaussian processes. That means that the PDF f_{Y(t)}(y) is also a standard Gaussian PDF. m_Y = 0 since m_X = 0, and the variance can be determined like this:

    Var(Y(t))=E\left\{ (Y(t)-m_Y)^2\right\}=E\left\{ \[Y(t)\]^2\right\}=E\left\{ (X(t+1)-X(t))^2\right\}=E\left\{(X(t+1))^2\right\}-2E\left\{ X(t+1)X(t)\right\}+E\left\{ (X(t))^2\right\}=2r_X(0)-2r_X(1)=1
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