Digital Signal Processing Reference
In-Depth Information
Chapter 3
Multidimensional Random Variables
3.1 What Is a Multidimensional Random Variable?
We have already explained that one random variable is obtained by mapping a
sample space
S
to the real
x-
axis. However, there are many problems in which the
outcomes of the random experiment need to be mapped from the space
S
to a two-
dimensional space (
x
1
,
x
2
), leading to a two-dimensional random variable (
X
1
,
X
2
).
Since the variables
X
1
and
X
2
are the result of the same experiment, then it is
necessary to make a joint characterization of these two random variables.
In general, the mapping of the outcomes from
S
to a
N
-dimensional space leads
to a
N
-dimensional random variable.
Fortunately, many problems in engineering can be solved by considering
only two random variables [PEE93, p. 101]. This is why we emphasize the two
random variables and the generalization of a two-variable case will lead to a
N
-dimensional case.
3.1.1 Two-Dimensional Random Variable
Consider the sample space
S
, as shown in Fig.
3.1
, and a two-dimensional space
where
x
1
and
x
2
are real axes,
1 < x
1
< 1
,
1 < x
2
< 1
. Mapping of the
outcome
s
i
to a point (
x
1
,
x
2
) in a two-dimensional space, leads to a two-dimensional
random variable (
X
1
,
X
2
).
If the sample space is discrete, the resulting two-dimensional random variable
will also be discrete. However, as in the case of a one variable, the continuous
sample space may result in continuous, discrete, or mixed two-dimensional random
variables. The range of a discrete two-dimensional random variable is made of
points in a two-dimensional space, while the range of a continuous two-dimensional
variable is a continuous area in a two-dimensional space. Similarly, a mixed
two-dimensional random variable has, aside from continuous area, additional
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