Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 191

3.5.3 Moment Theorem

Using the moment theorem for one random variable, as an analogy, we arrive at
the moment theorem that finds joint moments m nk

from the joint characteristic

function, as

o 1 ¼ 0

nþk f X 1 X 2 ð o 1 ; o 2 Þ
@o 1 n

nþk @

m nk ¼ðjÞ

o 2 ¼ 0 :

(3.209)

@o 2 k

3.5.4 PDF of the Sum of Independent Random Variables

The random variable Y is equal to the sum of the independent random variables
X 1 and X 2 ,

Y ¼ X 1 þ X 2 : (3.210)

The PDF of the variable Y (see ( 2.387 ) ) is expressed by its characteristic
function as,

1

1
2 p

f Y ðoÞ e joy d o:

f Y ðyÞ¼ (3.211)

1

Placing ( 3.201 ) into ( 3.211 ), we have:

1

1
2 p

f X 1 ðoÞf X 2 ðoÞ e joy d o:

f Y ðyÞ¼ (3.212)

1

By applying the relation ( 2.386 ) to the characteristic function f X 1 ðoÞ in ( 3.212 ),
and interchanging the order of the integrations, we arrive at:

1

f X 2 ðoÞ e joy d o 1
1

1
2 p

f X 1 ðx 1 Þ e jox 1 d x 1

f Y ðyÞ¼

1

1 1

1
2 p

f X 2 ðoÞ e joðyx 1 Þ d o: ¼ f X 1 ðx 1 Þ d x 1

(3.213)

1 1