Random Variable - Random Signals and Processes Primer with MATLAB - page 39

For the interval 1< x <a , we have:

PfX xg¼ 0 for 1< x <a: (2.29)

From ( 2.27 )to( 2.29 ), we arrive at:

8
<

0 for 1< x < 2 ;

x 2
4

F X ðxÞ¼ (2.30)

for 2 x 6 ;

:

1 for 6 < x <1:

This distribution is shown in Fig. 2.10b . Note that this distribution function
does not have jumps like the distribution of a discrete variable ( 2.24 ).
The distribution of a mixed random variable has one or more jumps and the
continuous part. To this end, it is possible to write the distribution of a mixed
random variable [THO71, pp. 50-51] as:

F X ðxÞ¼aF c ðxÞþF d ðxÞ; (2.31)

where F c is a distribution of the continuous variable and F d is a distribution of the
discrete random variable, and

0 a 1 ; (2.32)

a ¼ 1 X
i

PfX ¼ x i g: (2.33)

The proof can be found in [THO71, pp. 50-51].
A distribution where a ¼ 0 corresponds to a discrete r.v., while a distribution
where a ¼ 1 corresponds to a continuous r.v.
Example 2.2.4 In this example, we will illustrate the distribution of a mixed
random variable X ; which has one discrete value in x ¼ 2 with the probability
P {2} ¼ 1/3 and the continuous interval 2 < x < 6 (as shown in Fig. 2.11a ).

Fig. 2.11 Illustration of the distribution in Example 2.2.4