Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 257

Example 4.5.3 Find the PDF of the variable Y , where:

Y ¼ X

3

a i X i þ b i ; a 1 ¼ 2 ; a 2 ¼ 1 ; a 3 ¼ 1 : 5 b 1 ¼ b 2 ¼ 1 ;

;

(4.115)

i¼ 1

b 3 ¼ 4 X i ¼ Nð 1 ;

2 Þ:

;

Solution According to ( 4.113 ), the variable Y is also a normal random variable
with parameters:

m Y ¼ X

3

a i m i þ b i ¼ 2 1 þ 1 þð 1 Þ 1 þ 1 þ 1 : 5 1 þ 4 ¼ 8 : 5 : (4.116)

i¼ 1

s Y ¼ X

3

a i s i ¼ 4 2 þ 1 2 þ 2 : 25 2 ¼ 14 : 5 : (4.117)

i¼ 1

1
29 p

e ðy 8 : 5 Þ 2 f Y ðyÞ¼ p : (4.118)

29

4.5.4 Central Limit Theorem

The results presented in this chapter in relation to the sum of independent normal
variables can be viewed as a special case of the more general central limit theorem
(CLT).
According to the CLT, the sum of N independent random variables X i , each of
which contributes a small amount to the total, approaches the normal random
variable.

X ¼ X

N

X i : (4.119)

i¼ 1

The PDF of the variable X is a convolution of the PDFs of the individual
variables X i (see ( 3.218 )to( 3.219 )):

f X ðxÞ¼f X 1 ðxÞf X 2 ðxÞf X N ðxÞ:

(4.120)

According to ( 3.96 ), the mean value of the sum ( 4.119 ) is:

m X ¼ X

N

m X i : (4.121)

i¼ 1

Similarly, using the result ( 3.129 ) we find the variance of the variable X i in ( 4.119 ):

X ¼ X

N

s 2 s 2

X i :

(4.122)

i¼ 1