Random Variable - Random Signals and Processes Primer with MATLAB - page 33

Example 2.1.3 This experiment is a measurement of voltage and the outcomes are
continuous values:

S ¼fs j V 2 s V 1 g: (2.7)

The random variable X ( s ) is obtained by mapping the continuous space onto the
continuous range on the x -axis, as shown in Fig. 2.5 .
Note the following:

￿ The sample space for the continuous random variable must be continuous.
Otherwise, some values in the continuous range x would not have the corres-
ponding count-pairs in the sample space S .
￿ The continuous space does not always result in a continuous random variable.
Generally, from the continuous space, we can obtain the continuous random
variable, discrete random variable, and also mixed random variable as shown in
Examples 2.1.4 and 2.1.5.

Example 2.1.4 Consider the continuous space S from Example 2.1.3. In this
example, we are only interested in whether the voltage is positive or negative.
That is we have only two possibilities and as mentioned before, it is easiest to assign
the numerical values 0 and 1, as shown in Fig. 2.6 . Therefore, we can define the
random variable X ( s ) in the following way,

Xðs 0 Þ¼ 0 ;

Xðs < 0 Þ¼ 1 :

(2.8)

Note that the random variable is discrete and the sample space is continuous.
Note also that as opposed to Example 2.1.3, where the numerical values in the
x -axis correspond to the values of the voltage, in this example the numerical values
0 and 1 do not have anything to do with the physical values of the voltage in the
sample space S .

Fig. 2.5 The continuous random variable in Example 2.1.3