p. 61
Jointly Distributed Random Variables
• Recall
. In Chapters 4 and 5, focus on
univariate
random variable.
However, often a single experiment will have more than one
random variable which is of interest.
Ω
R
R
R
•
•
•
•
•
•
X
1
X
2
X
n
P
X
=(
X
1
, … ,
X
n
):
→
R
n
.
Definition. Given a sample space
and a probability measure
P
defined on the subsets of
, random variables
X
1
,
X
2
, … ,
X
n
:
→
R
are said to be
jointly distributed
.
We can regard
n
jointly distributed r.v.’s as a
random vector
Ω
•
•
•
•
•
•
X
1
X
2
P
A
E
A
P
X
1
,X
2
P
X
1
,X
2
(
A
)
=??
R
2
A
occurs
E
A
occurs
P
X
1
,X
2
(
A
)=
P
(
E
A
)
(
X
1
,
X
2
)
•
Q
: For
A
⊂
R
n
, how to define the probability of {
X
∈
A
} from
P
?
p. 62
For
A
⊂
R
n
,
For
A
i
⊂
R
,
i
=1, …,
n
,
P
X
1
,...,X
n
(
X
1
∈
A
1
,
···
,X
n
∈
A
n
)
=
P
(
{
ω
∈
Ω

X
1
(
ω
)
∈
A
1
}
∩
∩
{
ω
∈
Ω

X
n
(
ω
)
∈
A
n
}
)
P
X
1
,...,X
n
(
A
)
=
P
(
{
ω
∈
Ω

(
X
1
(
ω
)
,...,X
n
(
ω
))
∈
A
}
)
X
1
X
2
Definition. The probability measure of
X
(
P
X
, defined on
R
n
) is
called the
joint distribution
of
X
1
, …,
X
n
. The probability
measure of
X
i
(
, defined on
R
) is called the
marginal
distribution
of
X
i
.
P
X
i
•
Q
: Why need joint distribution? Why are marginal distributions
not enough?
Example (Coin Tossing, LNp.42).
X
2
: # of head
on 1
st
toss
X
1
: total # of heads
0
(1/8)
1
(3/8)
2
(3/8)
3
(1/8)
0
(1/2)
1/8
[
1/16
]
2/8
[
3/16
]
[
3/16
]
0
[
1/16
]
1
(1/2)
0
[
1/16
]
[
3/16
]
[
3/16
]
[
1/16
]