Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 175

Using the expression for the mean value of the function of two random variables
( 3.87 ), we arrive at:

1 1

k

n

m nk ¼ ðx 1 X 1 Þ ðx 2 X 2 Þ f X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.119)

1 1

This expression can be generalized for N random variables X 1 , , X N :

...

1 1

n N f X 1 ; ... ;X N ðx 1 ; ... ; x N Þ d x 1 ; ... ;

n 1

m n 1 ; ... ;n N ¼ ðx 1 X 1 Þ ; ... ; ðx N X N Þ d x N :

...

1 1

(3.120)

The second central moment m 11 , called covariance is especially important:

C X 1 X 2 ¼ m 11 ¼ E ðX 1 X 1 ÞðX 2 X 2 Þ

1 1

¼ ðx 1 X 1 Þðx 2 X 2 Þf X 1 X 2 ðx 1 ; x 2 Þ d x 1 d x 2 : (3.121)

1 1

Let us first relate the covariance to the independent variables , where the joint
PDF is equal to the product of the marginal PDFs, resulting in the following
covariance:

1 1

C X 1 X 2 ¼ m 11 ¼ ðx 1 X 1 Þðx 2 X 2 Þf X 1 ðx 1 Þf X 2 ðx 2 Þ d x 1 d x 2

1 1

2
4

3
5

2
4

3
5

1 1 1 1

¼ x 1 f X 1 ðx 1 Þ d x 1 X 1 f X 1 ðx 1 Þ d x 1 x 2 f X 2 ðx 2 Þ d x 2 X 2 f X 2 ðx 2 Þ d x 2

1 1 1 1

X 2 X 2 ¼ 0 :

¼ X 1 X 1

(3.122)

From ( 3.122 ) it follows that the covariance is equal to zero for the independent
variables.
Using ( 3.95 ), ( 3.121 ) can be simplified as:

C X 1 X 2 ¼ X 1 X 2 X 1 X 2 : (3.123)

Let us now relate covariance with the correlated and orthogonal random
variables.
From ( 3.123 ) and ( 3.111 ), it follows that the covariance is equal to zero if the
random variables are uncorrelated.