Suppose the random process Zt is defined as Zt X Yt where

Suppose the random process Z(t) is defined as Z(t) = X + Yt, where X and Y are independent random variables, where X is a random variable uniform on [0, 1] and Y is a random variable uniform on [0,2]. Compute the mean and autocorrelation of Z(t).

Solution

Z(t) = X +Yt where X ~ U[0,1] therefore E(x) = 1/2 and var (x) = 1/12 and Y ~ U[0,2] therefore E(x) = 1 and var (x) = 1/3

E (Z(t)) = E(x) + E(y)t = 0.5 + t

Corr (Z(t)) of lag t = Cov ( Z(s), Z(s+t)) / cov (Z(s), Z(s+t))

= Cov ( X+Ys, X+Y(t+s))/ sqrt(var(x)* var(y))

= {cov(x,x) + t cov(y,y)}/ sqrt(var(x)* var(y))

= [var(x) + tvar(y)]/ sqrt(var(x)* var(y))

= [1/12 + t(1/3)]/ sqrt(1/36)

= 1+4t/ (12 * 0.16667)

= (1+4t)/2

Corr(Z(t)) = 0.5 + 2t