STAT 4005 ASSIGNMENT 1 Due date: October 5, 2015 2 Let at N ID(0, a ). 1 1. Suppose E(X) = 3, V ar(X) = 2, E(Y ) = 0, V ar(Y ) = 4, and Corr(X, Y ) = 4 . 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 autocovariance function k . (a) Find the mean function and the autocovariance function of {Zt }. (b) Is {Zt } stationary? Why? 2 3. Let Zt = 0.4at + 0.5at1 + 0.6at2 + 0.7at3 + 0.8at4 with a = 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? 10 Zt ). (e) Find V ar( t=1 4. Suppose that Zt = (at + at1 + at2 + at3 )/4. Show that {Zt } is stationary and nd, k , 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 0.5Wt1 and Zt = Yt + 0.4Yt1 0.4Yt2 . Put Vt = Xt Zt . Find the Cov(Vt , Vtk ), k = 0, 1, 2, 3, .... 6. Let {Xt } be a zero-mean, unit-variance, stationary process with autocorrelation function k . Let Zt = 8 + 2t + 4tXt . (a) For {Zt }, nd the mean, variance, and autocovariance functions. (b) Is {Zt } stationary