Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 255

This result can be generalized for an arbitrary sum of N independent random
variables:

X ¼ X

N

X i ¼ Nðm i ; s i Þ:

X i ; (4.108)

i¼ 1

The variable X is also a normal variable Nðm; s 2 Þ with the parameters:

m ¼ X ¼ X

N N

s 2 s i :

m i ; (4.109)

i¼ 1 i¼ 1

Example 4.5.2 Find the PDF of the random variable X ,

X ¼ X 1 þ X 2 : (4.110)

where i ¼ 1, 2 are normal random variables: X 1 ¼ N (1, 5), X 2 ¼ N (2.6, 8).
Solution According to the observations in this section, the random variable X is
also a normal random variable with the parameters:

s 2

m ¼ 1 þ 2 : 6 ¼ 3 : 6 ; ¼ 5 þ 8 ¼ 13 : (4.111)

The corresponding signals and PDFs are shown in Fig. 4.17 .

4.5.3 Sum of Linear Transformations
of Independent Normal Random Variables

random variables X i ( m i , s i 2 ). The Consider independent normal linear

transformation

Y i ¼ a i X i þ b i (4.112)

results in independent normal random variables Y i . Their sum denoted as Y ,

Y ¼ X Y i ¼ X

N N

a i X i þ b i (4.113)

i¼ 1 i¼ 1

is also normal variable with parameters

m Y ¼ X

Y ¼ X

N N

s 2 a 2

i s 2

a i m i þ b i ;

i :

(4.114)

i¼ 1 i¼ 1