Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 176

Therefore, from summing the above statements it follows that the covariance
equals to zero if the random variables are either independent or dependent but
uncorrelated .
Additionally, from ( 3.112 ) it follows that if the random variables are orthogo-
nal, then the covariance is equal to the negative product of their mean values .

C X 1 X 2 ¼EfX 1 gEfX 2 g:

(3.124)

3.3.3.1 Variance of the Sum of Random Variables

Consider the sum of two random variables X 1 and X 2 , which is itself a random
variable X :

X ¼ X 1 þ X 2 : (3.125)

By applying the definition ( 2.333 ) of the variance of the random variable X and
using ( 3.88 ) and ( 3.125 ), we get:

2 2

s X ¼ ðX XÞ

¼ ðX 1 þ X 2 X 1 þ X 2 Þ

2 2

¼ ðX 1 X 1 Þ þ ðX 2 X 2 Þ þ 2 ðX 1 X 1 ÞðX 2 X 2 Þ: (3.126)

The first two terms in ( 3.126 ) are the corresponding variances of the variables X 1
and X 2 , respectively, while the third averaged product is the covariance.
Therefore, ( 3.126 ) reduces to:

s X ¼ s X 1 þ s X 2 þ 2 C X 1 X 2 : (3.127)

Equation ( 3.127 ) states that the variance of the sum of the variables X 1 and X 2 is
equal to the sum of the corresponding variances if their covariance is equal to zero
( i.e., the variables are either independent or uncorrelated ).
Therefore, if the random variables X 1 and X 2 are either independent or uncorre-
lated ðC X 1 X 2 ¼ 0 Þ , then

s X 1 þX 2 ¼ s X 1 þ s X 2 :

(3.128)

The result ( 3.128 ) can be generalized to the sum of N either independent or
uncorrelated variables X 1 , , X N :

...

X i ¼ X

N

s 2 s 2

X i :

(3.129)

P

N

i¼ 1

i¼ 1