Lecture 6
ORIE3500/5500 Summer2009 Chen
Two-dimensional random vectors and discrete
case
1 Two-dimensional random vectors
Random vectors are also known as jointly distributed random variables. Most
models require keeping track of more than one random variables at a time.
We construct such a model by considering several random variables together
X
= (
X
1
,X
2
,...,X
k
)
.
Let us begin with two dimensional random vectors.
The probability distribution of a two dimensional random variable is char-
acterized by its
joint cumulative distribution function
or
joint cdf
. It is de-
ﬁned as
F
X,Y
(
x,y
) =
P
[
X
≤
x,Y
≤
y
]
,
-∞
< x,y <
∞
.
It is common to denote intersection of events involving random variables
with ‘commas’. This means that in the deﬁnition above
P
[
X
≤
x,Y
≤
y
]
is another way of writing
P
([
X
≤
x
]
∩
[
Y
≤
y
]).
F
X,Y
(
x,y
) is thus the
probability of the region ‘south-west’ of the point (
x,y
).
Properties of joint cdf
Joint cdfs behave like ordinary cdf once we take into consideration the changes
necessary since this is on a higher dimensional space.
1. For all (
x,y
), 0
≤
F
X,Y
(
x,y
)
≤
1.
2. It satisﬁes
lim
x
→-∞
F
X,Y
(
x,y
) = 0
,
∀
y
and lim
y
→-∞
F
X,Y
(
x,y
) = 0
,
∀
x.
3. Since the probability of the sample space is 1, we have
lim
x
→∞
,y
→∞
F
X,Y
(
x,y
) = 1
.
1