1
Lecture 8
Multi-dimensional
distributions
z
z
One may check that the following function
are valid jcdf’s. That is they satisfy all the
properties that a jcdf must posess.
z
Example:
1.
0
otherwise
2.
0
otherwise
=
)
,
(
y
x
F
XY
0
,
0
),
1
)(
1
(
2
≥
≥
−
−
−
−
y
x
e
e
y
x
=
)
,
(
y
x
F
XY
5
3
,
1
0
,
2
3
≤
≤
≤
≤
−
y
x
y
x
Classification of Two-Dimensional
Random Vectors
z
In analogy to random variables
the
random vectors may be classified into
three categories:
1.
Jointly continuous random vectors,
These are random vectors the cdf’s of
which contain no jumps.
2.
Jointly discrete random vectors
These are random vectors that take only
discrete values, e.g. only integers
•
Instead of using a jcdf to describe the
distribution of such a vector we use the joint
probability function (jpmf), which is defined
as follows:
Definition 5
Consider ,
where
Then the jpmf of (X,Y) is defined as
Clearly, if
is a jcdf of (X,Y) and
is its jpmf, then
S
∈
ζ
,...}.
,...
{
)
(
,...},
,...
,
{
)
(
1
2
1
m
n
y
y
Y
x
x
x
X
∈
∈
,...
2
,
1
,
),
,
(
)
,
(
=
=
=
=
j
i
y
Y
x
X
P
j
i
P
j
i
XY
)
,
(
y
x
F
XY
)
,
(
j
i
P
XY
∑∑
==
=
]
[
1
]
[
1
),
,
(
)
,
(
x
i
y
j
XY
XY
j
i
P
y
x
F
)
,
(
Y
X
X
=