Stat1301B
Probability& Statistics I
Fall 2009-2010
P.104
Chapter III
Jointly Distributed Random Variables
§ 3.1
Joint and Marginal Distributions
When an experiment or survey is conducted, two or more random variables are
often observed simultaneously not only to study their individual probabilistic
behaviours but also to determine the degree of relationship among the variables as
in most of cases, the variables are related. The probabilistic behaviours of the
random variables are described by their
joint distribution
.
In the simplest case, suppose there are only two discrete random variables (
X
,
Y
)
which take distinct values :
Values of
X
:
(
)
}
,...,
,
{
2
1
r
x
x
x
X
=
Ω
Values of
Y
:
(
)
}
,...,
,
{
2
1
c
y
y
y
Y
=
Ω
Definition
The
joint probability mass function
(
joint pmf
) of the discrete random variables
X
and
Y
and is defined by
(
)
(
)
y
Y
x
X
P
y
x
p
=
=
=
,
,
,
(
)
Ω
∈
X
x
,
(
)
Ω
∈
Y
y
.
Sometimes the joint pmf can be conveniently presented in the form of a two-way
table as
Values of
Y
Values of
X
1
y
2
y
…
c
y
1
x
(
)
1
1
,
y
x
p
(
)
2
1
,
y
x
p
…
(
)
c
y
x
p
,
1
2
x
(
)
1
2
,
y
x
p
(
)
2
2
,
y
x
p
…
(
)
c
y
x
p
,
2
…
…
…
…
…
r
x
(
)
1
,
y
x
p
r
(
)
2
,
y
x
p
r
…
(
)
c
r
y
x
p
,

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