# Stochastic process, expected absolute deviation

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• September 10th 2011, 08:00 AM
Mondreus
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.
• September 12th 2011, 10:22 AM
Mondreus
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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