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?