Multidimensional Random Variables - Random Signals and Processes Primer with MATLAB - page 172

Equation ( 3.95 ) shows the following statement:

The mean value of the sum of two random variables is equal to the sum of the
corresponding mean values.

Note that no conditions have been imposed to obtain the result ( 3.95 ). A more
general result, which includes the N random variables, can be expressed as:

X

X k ¼ X

N N

X k : (3.96)

k¼ 1 k¼ 1

Example 3.3.2 Verify the relation ( 3.95 ) for the discrete random variables from
Example 3.3.1.

Solution From ( 3.80 ), we have:

X 1 þ X 2 ¼ X

X

2 2

j¼ 1 ðx 1 i þ x 2 j ÞPfX 1 ¼ x 1 i ; X 2 ¼x 2 j g¼ 3

= 4 1 = 4 ¼ 1 = 2 : (3.97)

i¼ 1

To verify the result ( 3.97 ), we found the following probabilities for random
variables X 1 and X 2 , using ( 1.67 ):

PfX 1 ¼ x 11 ¼ 1 g¼PfX 1 ¼ x 11 ¼ 1 ; X 2 ¼ x 21 ¼ 1 g

þ PfX 1 ¼ x 11 ¼ 1 ; X 2 ¼ x 22 ¼ 1 g¼ 3 = 8 : (3.98)

PfX 1 ¼ x 12 ¼ 1 g¼PfX 1 ¼ x 12 ¼ 1 ; X 2 ¼ x 21 ¼ 1 g

þ PfX 1 ¼ x 12 ¼ 1 ; X 2 ¼ x 22 ¼ 1 g¼ 5 = 8 : (3.99)

PfX 2 ¼ x 21 ¼ 1 g¼PfX 2 ¼ x 21 ¼ 1 ; X 1 ¼ x 11 ¼ 1 g

þ PfX 2 ¼ x 21 ¼ 1 ; X 1 ¼ x 12 ¼ 1 g¼ 3 = 8 : (3.100)

PfX 2 ¼ x 21 ¼ 1 g¼PfX 2 ¼ x 21 ¼ 1 ; X 1 ¼ x 11 ¼ 1 g

þ PfX 2 ¼ x 21 ¼ 1 ; X 1 ¼ x 12 ¼ 1 g¼ 5 = 8 : (3.101)

From ( 2.220 ) and ( 3.98 ) and ( 3.99 ), we have:

X 1 ¼ X

2

x 1 i PfX 1 ¼x 1 i g¼ 1 3 =

8 þ 1 5

=

8 ¼ 1

= :

4 (3.102)

i¼ 1

Similarly, from ( 3.100 ) and ( 3.101 ), it follows:

X 2 ¼ X

2

x 2 i PfX 2 ¼x 2 i g¼ 1 3 = 8 þ 1 5 = 8 ¼ 1 = 4 : (3.103)

i¼ 1