Let z1; z2 two random variables such that E[z1] = _1; E[z2] = _2; Var[z1] = _11; Var[z2] = _22; Cov[z1; z2] = _12, let the process x(t; !) be de_ned by x(t; !) = z1(w)IR􀀀[0(t) + z2(w)IR+(t). (a)...

Let z1; z2 two random variables such that E[z1] = _1; E[z2] = _2;

Var[z1] = _11; Var[z2] = _22; Cov[z1; z2] = _12, let the process x(t; !) be

de_ned by x(t; !) = z1(w)IR􀀀[0(t) + z2(w)IR+(t).

(a) Describe the trajectories of x.

(b) What should be necessary to make the process stationary?

(c) Calculate _x(t) and x(t1; t2).

(d) Find necessary and su_cient conditions on _1, _2, _1, _2 and _12

so that the process x is second order stationary. In this case, _nd

autocovariance and autocorrelation functions.