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ORIE 3500/5500 Fall Term 2008
Review 2
Multivariate Normal
A vector (
X
1
, . . . , X
k
) is said to follow a multivariate normal distribution
if any linear combination of the components, e.g.
Y
=
a
1
X
1
+
···
+
a
k
X
k
for some constants
a
1
, . . . , a
k
, has normal distribution. The parameters of
the multivariate normal distribution are (a) the mean vector
μ
and (b)
covariance matrix
Σ
μ
=
μ
1
μ
2
.
.
.
μ
k
,
Σ
=
σ
11
σ
12
···
σ
1
k
σ
21
σ
22
···
σ
2
k
.
.
.
.
.
.
.
.
.
σ
k
1
σ
k
2
···
σ
kk
,
and the notation is
X
=
X
1
X
2
.
.
.
X
k
∼
N
k
[
μ,
Σ
]
.
The interpretation of the parameters are
μ
i
=
E
(
X
i
)
, σ
ij
=
cpv
(
X
i
, X
j
)
,
1
≤
i, j
≤
k.
Three important things to remember:
1. Suppose
Y
= (
Y
1
, . . . , Y
r
)
T
be such that each com ponent of
Y
is some
linear cpmbination of the components of
X
, e.g.
Y
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This test prep was uploaded on 02/20/2009 for the course ORIE 3500 taught by Professor Weber during the Fall '08 term at Cornell University (Engineering School).
 Fall '08
 WEBER

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