1. In all the following questions {Zt} is a purely random process with mean E[Zt ] = 0, variance Var(Zt) = σ 2 , and successive values of Zt are independent so that Cov(Zt , Zt+k) = 0, k 6= 0.
(a) Derive the mean function of the process
Xt = Zt + 0.7Zt-1 - 0.2Zt-2.
Show that the autocorrelation function ρ(t) of {Xt} is given by
How many parameters does {Xt} have?
(b) Derive the mean and the autocorrelation functions of the process
How many parameters does {Xt} have?
(c) Consider the infinite-order process defined by
Xt = Zt + c(Zt-1 + Zt-2 + · · ·),
where c is a constant. Show that the process is not covariance-stationary. Also show that the series of first differences defined by
Yt = Xt - Xt-1
is covariance-stationary. Find the autocorrelation function ρ(t) of {Yt}. How many parameters does {Yt} have?
(d) Find the mean function µ(t) and the autocorrelation function ρ(t) of the process
Xt - µ = 0.7(Xt-1 - µ) + Zt .
Plot ρ(k) for k = -6, -5, . . . , -1, 0, +1, . . . , +6. How many parameters does {Xt} have?