Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 157

Let X 1 and X 2 indicate the occurrence of heads and tails for the first and second
coins, respectively.
The values of the random variable X 1 for the first coin tossing are x 1 ¼ 1 if heads
occurs, and x 1 ¼ 0, if tails occurs.
Similarly, for the variable X 2 , the value x 2 ¼ 1 indicates the occurrence of heads,
and x 2 ¼ 0 indicates tails in the second coin toss.
The mapping from the sample space S to the ( x 1 , x 2 ) space is shown in Fig. 3.4 .
Note that the values of the random variables are the same as in Example 3.1.1.
Therefore, one can obtain the same two-dimensional random variable, from differ-
ent random experiments.
The next example illustrates that, like in the one variable example, one can
obtain different two-dimensional random variables from the same experiment.

Example 3.1.3 Consider the same outcomes as in Example 3.1.1 for the following
mapping:

X 1 indicates at least one head occurs :

X 2 indicates at least one tail occurs : (3.5)

Therefore, the values of the random variable X 1 are x 1 ¼ 1, if at least one heads
occurs and x 1 ¼ 0 if no heads occurs. Similarly, the values of X 2 are x 2 ¼ 1ifat
least one tails occurs and x 2 ¼ 0 if no tails occurs. This mapping is shown in
Fig. 3.5 .
In a similar way, one can define a N -dimensional random variable by mapping
the outcomes of the space S to N -dimensional space, thus obtaining the N -dimen-
sional random variable,

ðX 1 ; X 2 ; ... ; X N Þ (3.6)

with the range:

ðx 1 ; x 2 ; ... ; x N Þ:

(3.7)

Fig. 3.4 Mapping the sample space S to ( x 1 , x 2 ) space in Example 3.1.2