1 HIGHER DIMENSIONAL RANDOM VECTORS
Lecture 8
ORIE3500/5500 Summer2009 Chen
Class Today
•
Random Vectors with higher dimensions
•
Independent Random Variables
1 Higher Dimensional Random Vectors
It is the expected extension of bivariate random vectors to higher dimensions.
(
X
1
,X
2
,...,X
n
) is said to be jointly distributed if each
X
i
is a random
variable and they have a joint cumulative distribution function
F
X
1
,
···
,X
n
(
x
1
,...,x
n
) =
P
[
X
1
≤
x
1
,...X
n
≤
x
n
]
.
Its properties are similar to that in the two-dimensional case, but it becomes
increasingly diﬃcult to write them down in mathematical notations. You
can try to write down the 4th property of joint cdfs for fun! It is important
to know how to get marginal distributions from the joint distribution. As
before, we get
F
X
1
(
x
1
) =
lim
x
i
→∞
,i
6
=1
F
X
1
,
···
,X
n
(
x
1
,...,x
n
) =
F
X
1
,
···
,X
n
(
x
1
,
∞
,...,
∞
)
.
More generally we can get the marginal of
X
k
as
F
X
k
(
x
k
) =
lim
x
i
→∞
,i
6
=
k
F
X
1
,
···
,X
n
(
x
1
,...,x
n
) =
F
X
1
,
···
,X
n
(
∞
,...,
∞
,x
k
,
∞
,...,
∞
)
.
We can get the marginal distributions of the random variables from their
joint cdf, but again we can not get full information on their joint behaviour
or joint cdf, just by knowing their marginals. Similarly we can also try to get
higher dimensional marginals from the joint cdf. The joint cdf of (
X
1
,X
2
)
can be obtained by
F
X
1
,X
2
(
x
1
,x
2
) =
F
X
1
,
···
,X
n
(
x
1
,x
2
,
∞
,...,
∞
)
.
One can get the joint the distribution of any group of random variables taken