Random Variable - Random Signals and Processes Primer with MATLAB - page 48

A mixed random variable has a continuous density within range where the
values are continuous and the delta functions in discrete points of x . From ( 2.31 ),
we write the PDF of a mixed random variable:
f X ðxÞ¼af c ðxÞþ X
i

PfX ¼ x i gdðx x i Þ;

(2.78)

where f c is the density in the continuous range of the mixed r.v. and
a ¼ 1 X
i

PfX ¼ x i g: (2.79)

The next example illustrates the density of the mixed random variable.

Example 2.3.3 Consider the mixed random variable from Example 2.2.4. Its
distribution is again plotted in Fig. 2.17 . Find its density as well as the probability
that the r.v. is less than 4.
Solution From ( 2.40 ), we find the density to be:

8
<

0 for x < 2 ;

1
3 dðx 2 Þþ

2
3

1
4

f X ðxÞ¼ ; (2.80)

for 2 x 6

:

0 for x > 6 :

The density is shown in Fig. 2.17 .
The desired probability is

d x ¼

ð ð

4 4

1
3 dðx 2 Þþ

1
6

1
3 þ

4 2
6 ¼

2
3 :

PfX < 4 g¼ f X ðxÞ d x ¼ (2.81)

2 2

This probability is illustrated in Fig. 2.17 , demonstrating that the probability is
the shaded area plus the area of the delta function. The probability is also illustrated
in the distribution plot as the value of the distribution in the point x ¼ 4.

Fig. 2.17 Illustration of PDF for a mixed r.v.