Page 1
Chapter 12
Multivariate normal distributions
The multivariate normal is the most useful, and most studied, of the standard joint dis
tributions in probability. A huge body of statistical theory depends on the properties of fam
ilies of random variables whose joint distribution is at least approximately multivariate nor
mal. The bivariate case (two variables) is the easiest to understand, because it requires a
minimum of notation; vector notation and matrix algebra becomes necessities when many
random variables are involved.
The general bivariate normal is often used to model pairs of dependent random vari
ables, such as : the height and weight of an individual; or (as an approximation) the score a
student gets on a final exam and the total score she gets on the problem sets; or the heights
of father and son; and so on. Many fancy statistical procedures implicitly require bivariate
(or multivariate, for more than two random variables) normality.
Bivariate normal
The most general bivariate normal can be built from a pair of independent random vari
ables,
X
and
Y
, each distributed
N
(
0
,
1
)
. For a constant
ρ
with

1
< ρ <
1, define random
variables
U
=
X
and
V
=
ρ
X
+
p
1

ρ
2
Y
That is,
(
U
,
V
)
=
(
X
,
Y
)
A
where
A
=
1
ρ
0
p
1

ρ
2
¶
Notice that
E
U
=
E
V
=
0,
var
(
V
)
=
ρ
2
var
(
X
)
+
(
1

ρ
2
)
var
(
Y
)
=
1
=
var
(
U
),
and
cov
(
U
,
V
)
=
ρ
cov
(
X
,
X
)
+
p
1

ρ
2
cov
(
X
,
Y
)
=
ρ.
Consequently,
correlation
(
U
,
V
)
=
cov
(
U
,
V
)/
p
var
(
U
)
var
(
V
)
=
ρ
From Chapter 10, the joint density for
(
U
,
V
)
is
1

det
A

f
(
(
u
, v)
A

1
)
,
where
f
(
x
,
y
)
=
1
2
π
exp

x
2
+
y
2
2
¶
all
x
,
y
The matrix
A
has determinant
p
1

ρ
2
and inverse
A

1
=
p
1

ρ
2

ρ
0
1
¶
/
p
1

ρ
2
Statistics 241: 16 November 1997
c David Pollard
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Chapter 12
Multivariate normal distributions
Page 2
If
(
x
,
y
)
=
(
u
, v)
A

1
then
x
2
+
y
2
=
(
u
, v)
A

1
(
A

1
)
0
(
u
, v)
0
=
(
u
, v)
1

ρ

ρ
0
¶
(
u
, v)
0
/(
1

ρ
2
)
=
u
2

2
ρ
u
v
+
v
2
1

ρ
2
Thus
U
and
V
have joint density
1
2
π
p
1

ρ
2
exp

u
2

2
ρ
u
v
+
v
2
2
(
1

ρ
2
)
¶
for all
u
, v.
The joint distribution is sometimes called the
standard bivariate normal
distribution
•
standard bivariate normal
with correlation
ρ
.
The symmetry of
ψ
in
u
and
v
implies that
V
has the same marginal distribution as
U
,
that is,
V
is also
N
(
0
,
1
)
distributed. The calculation of the marginals densities involves the
same integration for both variables.
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 Fall '11
 Staff
 Normal Distribution, Probability theory, probability density function, Multivariate normal distribution, David Pollard

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