Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 270

Exercise 4.10 The independent normal random variables X 1 and X 2 are related
with the random variable Y , as shown in Fig. 4.25 . Find the variance for the random
variable Y if the parameters of the variables X 1 and X 2 are m 1 ¼ 2, s 1 ¼ 4 and
m 2 ¼ 0, s 2 ¼ 5, respectively. (Use Q function.)
Answer The random variable X is a sum of independent normal random variables
and has the following parameters:

s 2 ¼ s 1 þ s 2 ¼ 9

;

m ¼ m 1 þ m 2 ¼ 2 :

The variable X is also a normal random variable, X¼ N (2, 9).
The output random variable Y is a discrete random variable with the discrete
values 2 and 0. The corresponding probabilities are:

PfY ¼ 2 g¼PfX< 8 g ; PfY ¼ 0 g¼PfX> 8 g: (4.185)

Using ( 4.53 ), we have:

PfY ¼ 2 g¼PfX 8 g¼ 1 PfX> 8 g¼ 1 QðkÞ

¼ 1 Qð 2 Þ;

j 8 2 j

¼ 1 Q

3

PfY ¼ 0 g¼ 1 PfY ¼ 2 g¼Qð 2 Þ: (4.186)

Using ( 4.186 ), the variance of the variable Y is obtained as:

Y 2

s Y ¼ Y 2

2 2

¼ 4 ð 1 Qð 2 ÞÞ ½ 2 ð 1 Qð 2 Þ ¼ 4 Qð 2 Þ 4 ½Qð 2 Þ : (4.187)

The function Q is related to the erf function as shown in Table 4.1 , resulting in:

2

4
2

2
2

4
2

2
2

s 2

p p

Y ¼

1 erf 1 erf

¼ 0 : 091 0 : 0041 ¼ 0 : 0869 : (4.188)

Fig. 4.25 Random variables X 1 , X 2 , and Y