Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 268

Exercise 4.7 A normal noise with a zero mean and a variance of s 2 enter into a
circuit with the characteristics shown in Fig. 4.23a . Find the expression and plot the
PDF of the noise in the circuit's output.

Answer For the negative values of the input noise the output noise will be equal to
zero,

PfY ¼ 0 g¼PfX< 0 g: (4.170)

The PDF of the input noise is symmetrical around zero, resulting in:

PfY ¼ 0 g¼PfX< 0 g¼ 1 = 2 : (4.171)

For the positive values of the input noise, the output noise Y will be equal to the
input noise thus having

y 2
2 s 2

1
2 ps 2

e

p

f Y ðyÞ¼f X ðxÞ¼ for y> 0 : (4.172)

From ( 4.170 )to( 4.172 ), we have:

8
<
:

=

2 dðyÞ

y ¼ 0

1 for

1
2 ps 2

e y 2

f Y ðyÞ¼ : (4.173)

p for y> 0

2 s 2

The density is shown in Fig. 4.23b .

Exercise 4.8 Find the characteristic function of the random variable Y at the output
of the block shown in Fig. 4.24 , if the input random variable X is the normal
variable with the zero mean value and the variance s 2 ¼ 1.

Answer The output random variable Y is a discrete random variable with the values
0 and 1.

PfY ¼ 1 g¼PfX< 1 : 5 : (4.174)

Fig. 4.23 The circuit characteristic and the output PDF