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Multivariate Distributions
Enrique CamposN´
a˜nez
The George Washington University
ApSc 116
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View Full Document Multivariate Data: Height, Weight, and Fat
Motivating multivariate distributions
Table of Contents
Bivariate Distributions
Conditional Distributions
Multivariate Distributions
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View Full Document Bivariate Distributions
I
The joint probability distribution of two r.v.’s is called a
bivariate
distribution
.
I
The distribution is discrete if
both
r.v.’s are discrete.
I
Deﬁnition:
The joint probability function (or joint p.f.) of
X
and
Y
is the function
f
such that
∀
(
x
,
y
)
∈ <
2
,
f
(
x
,
y
) =
Pr
(
X
=
x
,
Y
=
y
)
.
I
If (
x
,
y
) is not one of the possible pairs of values of (
X
,
Y
) then
f
(
x
,
y
) = 0. Always,
f
(
x
,
y
)
≥
0.
I
If the sequence
{
(
x
i
,
y
i
) :
i
= 1
,
2
, . . .
}
contains all possible values of
(
X
,
Y
), then
∞
X
i
=1
f
(
x
i
,
y
i
) = 1
.
I
Also,
Pr
±
(
X
,
Y
)
∈
A
⊂ <
2
²
=
X
(
x
i
,
y
i
)
∈
A
f
(
x
i
,
y
i
)
.
Exercise
Let
X
1
,
X
2
represent the number of points in two dice, and let
X
= min
{
X
1
,
X
2
}
,
Y
= max
{
X
1
,
X
2
}
.
What is the joint probability function of
X
and
Y
?
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View Full Document Continuous Bivariate Distributions
I
We say that
X
and
Y
have a
continuous joint distribution
if there is
a function
f
(
x
,
y
)
≥
0 that is deﬁned for all (
x
,
y
)
∈ <
2
and such
that
∀
A
⊂ <
2
,
Pr
[(
X
,
Y
)
∈
A
] =
Z
A
Z
f
(
x
,
y
)
dxdy
.
I
The function
f
(
x
,
y
) is the
joint probability density function
or joint
p.d.f., of
X
and
Y
.
I
In analogy to the univariate case,
f
(
x
,
y
) must satisfy
f
(
x
,
y
)
≥
0
,
∞
<
x
,
y
<
∞
,
and
Z
∞
∞
Z
∞
∞
f
(
x
,
y
)
dxdy
= 1
.
I
Note that if
X
and
Y
have a continuous joint distribution, then
1.
Any point, or inﬁnite sequence of points, in
<
2
has a probability 0,
and
2.
Any onedimensional curve in
<
2
has probability 0.
Mixed Bivariate Distributions
This covers all other cases.
Example
Select one person at random from members of the class and let
X
= 1 if
the person is male,
X
= 2 if the person is female, and let
Y
be the
weight in kilograms of the person.
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View Full Document Bivariate Distribution Functions I
Deﬁnition:
The
joint distribution function
, or joint d.f., of r.v.’s
X
and
Y
is the function
F
such that
∀
(
x
,
y
)
∈ <
2
,
F
(
x
,
y
)
≡
Pr
(
X
≤
x
,
Y
≤
y
)
.
Result.
For all
a
≤
b
, and
c
≤
d
, we have
Pr
(
a
<
X
≤
b
,
c
<
Y
≤
d
) =
F
(
b
,
d
)

F
(
a
,
d
)

F
(
b
,
c
) +
F
(
a
,
c
)
.
a
b
c
d
Bivariate Distribution Functions II
Deﬁnition.
The function
F
1
(
x
)
≡
Pr
(
X
≤
x
) = lim
y
→∞
Pr
(
X
≤
x
,
Y
≤
y
)
= lim
y
→∞
F
(
x
,
y
)
,
and is called the
marginal d.f.
of
X
.
Similarly,
F
2
(
y
)
≡
Pr
(
Y
≤
y
) = lim
x
→∞
F
(
x
,
y
)
,
and
F
2
(
y
) is called the
marginal d.f.
of
Y
.
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This note was uploaded on 01/02/2010 for the course APSC 116 taught by Professor Tommazzuchi during the Fall '08 term at GWU.
 Fall '08
 TomMazzuchi

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