Random Variable - Random Signals and Processes Primer with MATLAB - page 129

Fig. 2.62 PDF of the input variable and the characteristic of the quantizer. ( a ) PDF. ( b ) Quantizer

and

D ¼ a= 5 : (2.479)

Find the mean value of the random variable Y at the output of the quantizer and
plot both the PDF and the distribution.

Answer The output random variable Y is the discrete random variable, with the
discrete values:

y 0 ¼ 0 ; y 1 ¼ D; y 2 ¼ 2 D; y 3 ¼ 3 D; y 4 ¼ 4 D (2.480)

and the corresponding probabilities

PfY ¼ 0 g¼Pf 0 <X Dg¼ 1 = 5 ;

PfY ¼ Dg¼PfD<X 2 Dg¼ 1 = 5 ;

PfY ¼ 2 Dg¼Pf 2 D<X 3 Dg¼ 1 = ;

5

Pf Y ¼ 3 ;

PfY ¼ 4 Dg¼Pf 4 D<X 5 Dg¼ 1 = 5 :

Dg¼Pf 3 D<X 4 Dg¼ 1 =

5

(2.481)

The PDF and the distribution are shown in Fig. 2.63 .
The mean value is:

m Y ¼ X

4

1
5 D þ 2

y i Pfy i g¼ ½ D þ 3 D þ 4 D ¼ 2 D: (2.482)

i¼ 0

Note that the PDF of Y has the point of symmetry at 2 D , which is equal to the

mean value.

Exercise E.2.10 The PDF of the variable X is shown in Fig. 2.64a .

5e jxj f X ðxÞ¼ 0 : ; 1< x <1: (2.483)