Chapter 6: Jointly Distributed Random Variables
6-2
Example
6-1 (Roll two fair dice)
.
Let
X
be the smaller and
Y
be the larger outcome on
the dice. Write down the joint pmf of
X
and
Y
.
Sample space Ω =
{
(1
,
1)
,
(1
,
2)
,
· · ·
,
(6
,
6)
}
p
(
x, y
) =
P
(
X
=
x, Y
=
y
)
Example
6-2 (Roll two fair dice)
.
Let
X
be the sum of the two dice, and
Y
be the minimum.
Table the pmf function of
X
and
Y
.
Example
6-3
.
Two different balls drawn at random without replacement. An urn with 3
red, 4 white and 5 blue balls and we randomly select 3 balls from the urn. Let
X
and
Y
,
respectively, be the number of red and white balls chosen are randomly selected. Determine
the joint probability mass function of
X
and
Y
.
Joint distribution function of
X
and
Y
For any two RVs
X
and
Y
, the joint (cumulative) distribution function of
X
and
Y
is
defined by
F
X,Y
(
x, y
) =
P
{
X
≤
x, Y
≤
y
}
,
-∞
< x, y <
∞
In the continuous case, this is
Z
x
-∞
Z
y
-∞
f
X,Y
(
x, y
)
dy dx,
and so we have
f
(
x, y
) =
∂
2
∂x∂y
F
(
x, y
)
.
where
f
(
x, y
) is called the
(joint pdf)
of
X
and
Y
. (graphic & properties)
Probabilities of events determined by
X
and
Y
P
(
a
≤
X
≤
b, c
≤
Y
≤
d
) =
Z
b
a
Z
d
c
f
X,Y
(
x, y
)
dy dx,
or
P
((
X, Y
)
∈
D
) =
Z Z
D
f
X,Y
dy dx