AMS 507, Fall Semester, 2010
Chapter Six
Jointly Distributed Random Variables
6.1. Joint Distribution Functions
The
joint cumulative probability distribution function
of any two random variables
X
and
Y
is defined by
.
,
},
,
{
)
,
(
,
∞
<
<
∞

≤
≤
=
b
a
b
Y
a
X
P
b
a
F
Y
X
The cumulative distribution
function of
X
is related to the joint cumulative probability distribution function by
),
,
(
)
(
,
∞
=
a
F
a
F
Y
X
X
and the cumulative distribution function of
Y
is related to the joint
cumulative probability distribution function by
).
,
(
)
(
,
b
F
b
F
Y
X
Y
∞
=
In this context,
)
(
a
F
X
and
)
(
b
F
Y
are referred to as the
marginal distribution functions
of
X
and
Y
respectively. The probability of a rectangular region is given by
).
,
(
)
,
(
)
,
(
)
,
(
}
,
{
1
1
1
2
,
2
1
,
2
2
,
2
1
2
1
b
a
F
b
a
F
b
a
F
b
a
F
b
Y
b
a
X
a
P
Y
X
Y
X
Y
X
+


=
≤
<
≤
<
Definition of
joint probability mass function
of two discrete random variables:
p
x
y
P
X
x
Y
y
(
,
)
(
,
) .
=
=
=
One can calculate the marginal cumulative probability
function of
X
as well as the marginal cumulative probability function of