# STATS – In all the following questions {Zt} is a purely random process with mean

August 30, 2017

Question
1. In all the following questions {Zt} is a purely random process with mean E[Zt
] =
0, variance Var(Zt) = ?
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 ?(t) of {Xt} is given by
?(t) =
?
????
????
1 k = 0
0.37 k = ±1
?0.13 k = ±2
0 otherwise.
(1)
How many parameters does {Xt} have?
(b) Derive the mean and the autocorrelation functions of the process
Xt =
Xm
k=0
(m + 1)?1Zt?k.
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 ?(t) of {Yt}. How
many parameters does {Yt} have?
(d) Find the mean function µ(t) and the autocorrelation function ?(t) of the
process
Xt ? µ = 0.7(Xt?1 ? µ) + Zt
.
Plot ?(k) for k = ?6, ?5, . . . , ?1, 0, +1, . . . , +6. How many parameters does
{Xt} have?
1

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