Stat 430/Math468 – Notes #7
Chapter 4
The Multivariate Normal Distribution (Continued)
If a random vector
has multivariate normal distribution, then it has the following
properties:
X
1.
Linear combinations of the components of
are normally distributed.
X
2.
All subsets of the components of
have (multivariate) normal distribution.
X
3.
Uncorrelated components are independent.
4.
The conditional distributions of the components are (multivariate) normal.
Formally, we have the following results:
Result:
1.
If
and
, then
.
~(
,
p
N
X
μΣ
)
1
( ,.
..,
)'
p
aa
=
a
'~(
'
,
' )
N
aX
a
μ
a
Σ
a
2.
If
for
every
'
,
N
a
μ
a
Σ
a
1
( ,.
..,
p
=
a
, then
.
,
p
N
X
)
p
The proof needs multivariate moment generating function (skipped). The two results
together can be considered as a characteristic (definition) of multivariate normal
distribution.
Example: Let
, then what is
and what is its
distribution?
The ith element is 1 others are 0's
(
0,0,.
..0,1,0.
..,0
)',
1,2,.
..,
i
i
==
a
1442 4 43
'
i
1
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Suppose
.
~(
,
p
N
X
μΣ
)
)
)
1.
If
, then
.
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 Spring '10
 Hua
 Statistics, Normal Distribution, Multivariate normal distribution

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