168
Chapter 5
Multiple Random Variables
is given by
F
XY
(
x
,
y
)
=
P
[
X
≤
x
,
Y
≤
y
]
The pair
(
X
,
Y
)
is referred to as a
bivariate
random variable. If we define
F
X
(
x
)
=
P
[
X
≤
x
]
as the
marginal
CDF of
X
and
F
Y
(
y
)
=
P
[
Y
≤
y
]
as the
marginal
CDF
of
Y
, then we define the random variables
X
and
Y
to be independent if
F
XY
(
x
,
y
)
=
F
X
(
x
)
F
Y
(
y
)
for every value of
x
and
y
.
5.2.1
Properties of the Joint CDF
As a probability function
F
XY
(
x
,
y
)
has certain properties, which include the fol-
lowing:
1.
Since
F
XY
(
x
,
y
)
is a probability, 0
≤
F
XY
(
x
,
y
)
≤
1 for
−∞
<
x
<
∞
,
−∞
<
y
<
∞
.
2.
If
x
1
≤
x
2
and
y
1
≤
y
2
, then
F
XY
(
x
1
,
y
1
)
≤
F
XY
(
x
2
,
y
1
)
≤
F
XY
(
x
2
,
y
2
)
. Simi-
larly,
F
XY
(
x
1
,
y
1
)
≤
F
XY
(
x
1
,
y
2
)
≤
F
XY
(
x
2
,
y
2
)
. This follows from the fact that
F
XY
(
x
,
y
)
is a nondecreasing function of
x
and
y
.
3.
lim
x
→∞
y
→∞
F
XY
(
x
,
y
)
=
F
XY
(
∞
,
∞
)
=
1
4.
lim
x
→−∞
F
XY
(
x
,
y
)
=
F
XY
(
−∞
,
y
)
=
0
5.
lim
y
→−∞
F
XY
(
x
,
y
)
=
F
XY
(
x
,
−∞
)
=
0
6.
lim
x
→
a
+
F
XY
(
x
,
y
)
=
F
XY
(
a
,
y
)
7.
lim
y
→
b
+
F
XY
(
x
,
y
)
=
F
XY
(
x
,
b
)
8.
P
[
x
1
<
X
≤
x
2
,
Y
≤
y
] =
F
XY
(
x
2
,
y
)
−
F
XY
(
x
1
,
y
)
9.
P
[
X
≤
x
,
y
1
<
Y
≤
y
2
] =
F
XY
(
x
,
y
2
)
−
F
XY
(
x
,
y
1
)
10.
If
x
1
≤
x
2
and
y
1
≤
y
2
, then
P
[
x
1
<
X
≤
x
2
,
y
1
<
Y
≤
y
2
] =
F
XY
(
x
2
,
y
2
)
−
F
XY
(
x
1
,
y
2
)
−
F
XY
(
x
2
,
y
1
)
+
F
XY
(
x
1
,
y
1
)
≥
0
Also, the marginal CDFs are obtained as follows:
F
X
(
x
)
=
F
XY
(
x
,
∞
)
F
Y
(
y
)
=
F
XY
(
∞
,
y
)
From the above properties we can answer questions about
X
and
Y
.