Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 267

The density function of the output random variable is given as:

f Y ðyÞ¼ 0 : 5 dðy UÞþ 0 : 5 dðy þ UÞ; (4.166)

and it is shown in Fig. 4.21b .
Exercise 4.6 Find the PDF of the output random variable Y where the input
random variable X ¼ N (0, 1) and the output variable is obtained from the output
of the limiter, as shown in Fig. 4.22a .

Answer The output random variable Y is mixed, having two discrete values U 0 and
U 0 , and is continuous for { U 0 < X < U 0 }. The discrete values have the fol-
lowing probabilities:

h i

p 2

PfY ¼ U 0 g¼PfY ¼U 0 g¼ 0 : 51 þ erf ðU 0 = Þ : (4.167)

For the values of the input variable { U 0 < X < U 0 }, the output random
variable Y is equal to the input variable X , thus having the same PDF:

1
2 p

e y 2

p

f Y ðyÞ¼f X ðxÞ¼ for U 0 <Y<U 0 : (4.168)

From ( 4.167 ) and ( 4.168 ), the corresponding PDF of the variable Y is:

8
<
:

PfY ¼U 0 Þdðy þ U 0 Þ for Y ¼U 0

e y 2

f Y ðyÞ¼

1

2 p

: (4.169)

for U 0 <Y<U 0

p

PfY ¼ U 0 Þdðy U 0 Þ for Y ¼ U 0

The PDF ( 4.169 ) is shown in Fig. 4.22b .

Fig. 4.22 Transformation of a normal variable and the output density function