Stat 5101 (Geyer) Fall 2009
Homework Assignment 5
Due Wednesday, October 21, 2009
Solve each problem. Explain your reasoning. No credit for answers with
no explanation. If the problem is a proof, then you need words as well as
formulas. Explain why your formulas follow one from another.
51.
Suppose
X
1
,
X
2
,
. . .
are IID random variables having mean
μ
and
variance
τ
2
. For each
i
≥
1 define
Y
i
=
5
X
j
=1
X
i
+
j
Then
Y
1
,
Y
2
,
. . .
is called a
moving average of order 5
time series, MA(5) for
short. It is a weakly stationary time series. (So far this repeats the setup
for Problem 46.) If
Y
n
=
1
n
n
X
i
=1
Y
i
show that
Y
n

5
μ
=
O
p
(
n

1
/
2
)
.
52.
Suppose that
X
1
,
X
2
,
. . .
are IID random variables. For some subset
A
of the real numbers, define
Y
i
=
I
A
(
X
i
)
,
i
= 1
,
2
, . . .
and
Y
n
=
1
n
n
X
i
=1
Y
i
.
Show that
Y
n
converges in probability to Pr(
X
i
∈
A
), and in fact
Y
n

Pr(
X
i
∈
A
) =
O
p
(
n

1
/
2
)
.
Also show that
Y
n
is the fraction of
X
1
,
. . .
,
X
n
that lie in
A
.
53.
Define
Ω =
{
1
,
2
, . . . ,
10
}
A
=
{
ω
∈
Ω :
ω <
5
}
B
=
{
ω
∈
Ω : 2
< ω <
8
}
C
=
{
10
}
1
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(a) Calculate
A
c
where complements are taken relative to Ω.
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 Fall '02
 Staff
 Probability theory, Yi, iid random variables

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