Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 190

If the random variables X 1 and X 2 are independent, then the joint characteristic
function of their sum is equal to the product of the marginal characteristic functions,

f Y ðoÞ¼f X 1 ðoÞf X 2 ðoÞ: (3.201)

This result can be applied for the sum of N independent random variables X i ,

Y ¼ X

N

X i : (3.202)

i¼ 1

The joint characteristic function of ( 3.202 )is

f Y ðoÞ¼ Y

N

f X i ðoÞ: (3.203)

i¼ 1

Example 3.5.1 Find the characteristic function of the random variable Y , where

Y ¼ aX 1 þ bX 2 (3.204)

and where X 1 , and X 2 are independent.
Solution Using (2.386) and the definition of the characteristic function ( 3.194 ) and
the condition of independence ( 3.201 ), we have:

f Y ðoÞ¼Ef e joðaX 1 þbX 2 Þ g¼Ef e joaX 1 gEf e jobX 2

g¼f X 1 ðaoÞf X 2 ðboÞ:

(3.205)

Example 3.5.2 The random variables X 1 and X 2 are independent. Find the joint
characteristic function of the variables X and Y , as given in the following equations:

X ¼ X 1 þ 2 X 2 ;
Y ¼ 2 X 1 þ X 2 : (3.206)

Solution From the definition ( 3.194 ), and using ( 3.206 ), we have:

f XY ðo 1 ; o 2 Þ¼Ef e jðo 1 Xþo 2 Y g¼Ef e jo 1 ðX 1 þ 2 X 2 Þþjo 2 ð 2 X 1 þX 2 Þ g

¼ Ef e jX 1 ðo 1 þ 2 o 2 ÞþjX 2 ð 2 o 1 þo 2 Þ g: (3.207)

Knowing that the random variables X 1 and X 2 are independent from ( 3.207 ) and
( 3.201 ), we arrive at:

f XY ðo 1 ; o 2 Þ¼Ef e jX 1 ðo 1 þ 2 o 2 Þ gEf e jX 2 ð 2 o 1 þo 2 Þ g
¼ f X 1 ðo 1 þ 2 o 2 Þf X 2 ð 2 o 1 þ o 2 Þ:

(3.208)