Random Variable - Random Signals and Processes Primer with MATLAB - page 125

However, the r.v. X is less than 3 if it is equal to either 1 or 2, resulting in:

PfX < 3 g¼ 0 : 25 þ 0 : 25 ¼ 0 : 5 : (2.460)

This probability corresponds to the value of the distribution in the distribution
plot for x ¼ 3 and to the area below delta functions from 1 until 3 in the PDF
plot, i.e., again equal to 0.5.
Exercise E.2.5 The PDF of the random variable X is given in Fig. 2.59a . Find the
type of the random variable and plot the corresponding distribution function. Find
the probability that the random variable is equal to 1 and to 2, as well. Additionally,
find the probability that the random variable is greater than 1.

Answer The random variable is mixed (i.e., it is continuous in the interval [0, 2]
and has one discrete value X ¼ 2, indicated by the delta function in point 2 of the
PDF plot). The corresponding distribution function is the integral of the PDF, as
demonstrated in Fig. 2.59b .
The r.v is continuous in the interval [0, 2] and thus it does not take any particular
value in this interval, resulting in:

PfX ¼ 1 g¼ 0 : (2.461)

However, the random variable takes the discrete value 2 with the probability
equal to 1/4

PfX ¼ 2 g¼ 1 = 4 : (2.462)

Similarly, we have:

ð

2

3
8 þ

1
4 ¼

5
8 :

PfX> 1 g¼ f X ðxÞ d x ¼ (2.463)

1

Fig. 2.59 ( a ) PDF and ( b ) distribution of the mixed random variable