Stat 351 Fall 2007
Assignment #9 Solutions
1. (a)
By Definition I, we see that
X
1

ρX
2
is normally distributed with mean
E
(
X
1

ρX
2
) =
E
(
X
1
)

ρE
(
X
2
) = 0
and variance
var(
X
1

ρX
2
) = var(
X
1
) +
ρ
2
var(
X
2
)

2
ρ
cov(
X
1
, X
2
) = 1 +
ρ
2

2
ρ
2
= 1

ρ
2
.
That is,
X
1

ρX
2
=
Y
where
Y
∈
N
(0
,
1

ρ
2
). Hence,
Y
=
1

ρ
2
Z
where
Z
∈
N
(0
,
1). In
other words, there exists a
Z
∈
N
(0
,
1) such that
X
1

ρX
2
=
1

ρ
2
Z.
1. (b)
Since
X
= (
X
1
, X
2
) is MVN, and since
Z
=
X
1
1

ρ
2

ρX
2
1

ρ
2
,
we conclude that (
Z, X
2
) is also a MVN. Hence, we know from Theorem V.7.1 that the components
of a MVN are independent if and only if they are uncorrelated. We find
cov(
Z, X
2
) = cov
X
1
1

ρ
2

ρX
2
1

ρ
2
, X
2
=
1
1

ρ
2
cov(
X
1
, X
2
)

ρ
1

ρ
2
var(
X
2
)
=
ρ
1

ρ
2

ρ
1

ρ
2
= 0
which verifies that
Z
and
X
2
are, in fact, independent.
Exercise 4.2, page 127
: If
φ
(
t, u
) = exp
{
it

2
t
2

u
2

tu
}
= exp
{
it

1
2
(4
t
2
+ 2
tu
+ 2
u
2
)
}
, then
we recognize this as the characteristic function of a normal random variable
X
= (
X
1
, X
2
)
∈
N
1
0
,
4
1
1
2
.
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 Fall '08
 MichaelKozdron
 Normal Distribution, Probability, Variance, Probability theory, probability density function, X1

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