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ST561
Fall 2010
Homework 9
Due: Monday, Nov. 29th, 2010
1. Let
X
= (
X
1
,...,X
p
)
T
follow a multivariate normal distribution with mean
μ
and
covariance matrix Σ. That is
X
∼
MV N
(
μ
,
Σ) with joint pdf
f
X
(
x
) =
1
(2
π
)
p/
2

Σ

1
/
2
exp
'

(
x

μ
)
T
Σ

1
(
x

μ
)
/
2
“
where

Σ

is the determinant of Σ.
Let
Y
=
A
X
, where
A
is a
p
×
p
matrix
with its inverse
A

1
exists.
Use the method of transformation, show that
Y
∼
MV N
(
A
μ
,
A
Σ
A
T
).
2. Let
Z
= (
Z
1
,...,Z
p
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Unformatted text preview: ) T with { Z i } p i =1 being i.i.d. N (0 , 1). For any given positive deﬁnite p × p matrix Σ, using the result from (1), ﬁnd a transform matrix A such that Y = A Z follows a multivariate normal with mean μ = (0 ,..., 0) T and covariance Σ. 3. Textbook page 233, 4.4.8. 4. Textbook Page 233, 4.4.10. 1...
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This note was uploaded on 12/04/2010 for the course ST 561 taught by Professor Staff during the Fall '08 term at Oregon State.
 Fall '08
 Staff

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