Utah State University
ECE 6010
Stochastic Processes
Homework # 4 Solutions
1. Suppose
X
∼ N
(
μ
,
Σ).
(a) Show that
E
[
X
] =
μ
and cov(
X
,
X
) = Σ.
Using characteristic functions:
φ
X
(
u
) = exp[
i
u
T
μ

1
2
u
T
Σ
u
]
Taking the gradient with respect to
u
we have
∂φ
X
(
u
)
∂
u
= exp[
i
u
T
μ

1
2
μ
T
Σ
μ
](
i
μ

Σ
u
)
Now evaluating at
u
=
0
and dividing by
i
we obtain
E
[
X
] =
μ
.
Taking the gradient again with respect to
u
we have
exp[
i
u
T
μ

1
2
μ
T
Σ
μ
](

Σ) + exp[
i
u
T
μ

1
2
μ
T
Σ
μ
](
i
μ

Σ
u
)(
i
μ

Σ
u
)
T
and evaluating at
u
=
0
we have

Σ

μμ
Dividing by
i

2
we obtain
E
[
X
2
] = Σ+
μμ
T
, from which it follows that cov(
X
,
X
) =
Σ.
(b) Show that
A
X
+
b
∼ N
(
A
μ
+
b
, A
Σ
A
T
).
We note that
φ
X
(
u
) =
E
[
e
i
u
T
X
] = exp[
i
u
T
μ

1
2
μ
T
Σ
μ
]. Then
φ
Y
(
u
) =
E
[
e
i
u
T
Y
] =
E
[
e
i
u
T
(
A
X
+
b
)
] =
e
i
u
T
b
E
[
e
i
u
T
A
X
]
But we can compute the expected value using what we know from
φ
X
(
u
):
E
[
e
i
u
T
A
X
] =
φ
X
(
v
)

v
=
A
T
u
= exp[
i
u
T
A
μ

1
2
u
T
A
Σ
A
T
u
]
1
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So
φ
Y
(
u
) =
e
i
u
T
b
exp[
i
u
T
A
μ

1
2
u
T
A
Σ
A
T
u
] = exp[
i
u
(
A
μ
+
b
)

1
2
u
T
(
A
Σ
A
T
)
u
]
.
From the form of the characteristic function we identify that
Y
is Gaussian with
E
[
Y
] =
A
μ
+
b
cov(
Y
,
Y
) =
A
Σ
A
T
.
(c) Suppose Σ
>
0 and write Σ =
CC
T
. Show that
C

1
(
X

μ
)
∼ N
(
0
, I
).
If
Y
=
C

1
X

C

1
μ
we have, by applying the previous result
E
[
Y
] =
C

1
μ

C

1
μ
=
0
cov(
Y
,
Y
) =
C

1
Σ
C

T
=
C

1
(
CC
T
)
C

T
=
I.
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 Spring '08
 Stites,M
 Randomness, σx σy, Y, fX Y

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