Normal Random Variable - Random Signals and Processes Primer with MATLAB - page 254

2 =9

m=1, s

1

2 =36

m=0, s

0.8

0.6

0.4

0.2

0

1.5 1 0.5

- - - 0

w

0.5 1 1.5

Fig. 4.16 Characteristic functions for normal variables X and Y (absolute values)

The mean value and the variance of variable X are shown below (see ( 3.95 ) and
( 3.128 )):

s 2 ¼ s 1 þ s 2 : m ¼ m 1 þ m 2 ; (4.104)

The characteristic function of the sum of independent variables is equal to the
product of the characteristic functions:

fðoÞ ¼ f 1 ðoÞf 2 ðoÞ: (4.105)

The characteristic functions of the normal variables X i , i ¼ 1, 2, according to
( 4.100 ), is:

f X i ðoÞ ¼ e jom i o 2 s i 2 ; i ¼ 1 ; 2 : (4.106)

2

From ( 4.103 ) to ( 4.106 ) we have:

f X ðoÞ ¼ e jom 1 o 2 s 1 2

2 e jom 2 o 2 s 2 2

¼ e joðm 1 þm 2 Þ o 2 ð s 1 2 þ s 2 2 Þ ¼ e jom o 2 s 2 : (4.107)

2 2 2

Comparing ( 4.107 ) with the expression for the characteristic function of a
normal variable ( 4.100 ), we can see that the expressions are equal, thus indicating
that the sum of normal random variables is also a normal random variable.