# Thread: Stochastic process, expected absolute deviation

1. ## Stochastic process, expected absolute deviation

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

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

I have no idea where to begin except maybe setting $\displaystyle Y(t)=X(t+1)-X(t)$, but then I have to figure out what the PDF $\displaystyle f_{Y(t)}(y)$ is and I'm not sure how to proceed.

2. ## 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 $\displaystyle f_{Y(t)}(y)$ is also a standard Gaussian PDF. $\displaystyle m_Y = 0$ since $\displaystyle m_X = 0$, and the variance can be determined like this:

$\displaystyle 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$