1.    Suppose E(X) = 3, V ar(X) = 2, E(Y ) = 0, V ar(Y ) = 4, and Corr(X, Y ) = 1 .

Find (i) V ar(2X + Y ), (ii) Cov(Y, X + Y ), and (iii) Corr(X + Y, 2Y - X).

2.    Suppose Zt = 8 + 2t + 5Xt, where {Xt} is a zero-mean stationary series with auto covariance function γk .

(a)     Find the mean function and the auto covariance function of {Zt}.

(b)     Is {Zt} stationary? Why?

3.  Let Zt = 0.4at + 0.5at  1  + 0.6a  +  0.7a  +  0.8a  with σ2  = 1.

(a) Find V ar(Zt).

(b) Find Cov(Zt, Zt+k ), k = 0, 1, 2, ....

(c)      Find Corr(Zt, Zt+k ), k = 0, 1, 2, ....

(d)     Is {Zt} (weakly) stationary?

(e)     Find V ar(.Zt). t=1

4.    Suppose that Zt = (at + at-1 + at-2 + at-3)/4. Show that {Zt} is stationary and find, ρk, k = 0, 1, 2, 3, ....

5.     Suppose {Wt} and {Yt} are two independent normal white noise series with V ar(Wt) = 2V ar(Yt) =  4.   Let Xt  =  Wt - 0.5Wt-1  and Zt  =  Yt + 0.4Yt-1  - 0.4Yt-2.   Put

Vt = Xt - Zt. Find the Cov(Vt, Vt-k ), k = 0, 1, 2, 3, ....

6.    Let {Xt} be a zero-mean, unit-variance, stationary process with autocorrelation function ρk . Let

Zt = 8 + 2t + 4tXt.

(a)     For {Zt}, find the mean, variance, and auto covariance functions.

(b)     Is {Zt} stationary?