STAT 4005 ASSIGNMENT 1 | Q5 AND q6 Solution

STAT 4005 ASSIGNMENT 1
Due date: October 5, 2015

Let at NID(0; 2
a).
1. Suppose E(X) = 3; V ar(X) = 2; E(Y ) = 0; V ar(Y ) = 4, and Corr(X; Y ) = 1
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 fXtg is a zero-mean stationary series with
autocovariance function
k.
(a) Find the mean function and the autocovariance function of fZtg.
(b) Is fZtg stationary? Why?
3. Let Zt = 0:4at + 0:5at􀀀1 + 0:6at􀀀2 + 0:7at􀀀3 + 0:8at􀀀4 with 2
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 fZtg (weakly) stationary?
(e) Find V ar(
X10
t=1
Zt).
4. Suppose that Zt = (at + at􀀀1 + at􀀀2 + at􀀀3)=4. Show that fZtg is stationary and
nd, k; k = 0; 1; 2; 3; :::.
5. Suppose fWtg and fYtg 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 fXtg be a zero-mean, unit-variance, stationary process with autocorrelation
function k. Let
Zt = 8 + 2t + 4tXt:
(a) For fZtg, nd the mean, variance, and autocovariance functions.
(b) Is fZtg stationary?