Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 173

From ( 3.95 ), ( 3.97 ), ( 3.102 ), and ( 3.103 ), we arrive at:

X 1 þ X 2 ¼ 1 = 2 ¼ X 1 þ X 2 ¼ 1 = 4 þ 1 = 4 ¼ 1 = 2 : (3.104)

3.3.2 Joint Moments Around the Origin

The expected value of joint random variables X 1 and X 2 ,
EX 1 X 2 : (3.105)

is called the joint moment m r of the order r , where

r ¼ n þ k: (3.106)

Equation ( 3.105 ) presents the expected value of the function g ( X 1 , X 2 ) of the
random variables X 1 and X 2 , and thus can be obtained using ( 3.87 ):

1 1

m r ¼ EfX 1 X 2 g¼ x 1 x 2 f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.107)

1 1

This result can be generalized for N random variables X 1 , , X N in order to

...

obtain the joint moment around the origin of the order

r ¼ X

N

n i ;

(3.108)

i¼ 1

1 1

¼

EX n 1
1

; ... ; X n N x n 1 ; ... ;x n N f X 1 ; ... ;X N ðx 1 ; ... ; x N Þ d x 1 ; ... ; d x N : (3.109)

...

1 1

Special importance has been placed on the second moment, where n ¼ k ¼ 1,
which is called the correlation and has its proper denotation R X 1 X 2 ,

1 1

R X 1 X 2 ¼ EfX 1 X 2 g¼ x 1 x 2 f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.110)

1 1

Some important relations between variables X 1 and X 2 can be expressed using
the correlation.
For example, if the correlation can be written as a product of the expected values
of X 1 and X 2 ,

EfX 1 ; X 2 g¼EfX 1 gEfX 2 g; (3.111)

then it is said that the variables are uncorrelated .