Random Variable - Random Signals and Processes Primer with MATLAB - page 81

However, the sum of ( 2.227 ) does not converge

1

1
k !1;

(2.228)

k¼ 0

and consequently, the mean value of the random variable X does not exist.
If we consider an arbitrary function of x i , g ( x i ), instead of x i , the expression
( 2.214 ) becomes:

¼ X

N

N 1 g ð x 1 Þþþ N N g ð x n Þ
M

N i
M gðx i Þ:

m empg ¼ (2.229)

i¼ 1

For a high M value ( M >> 1), the expression ( 2.229 ) becomes the expected value

of g ( X ):

EfgðXÞg ¼ X

N

gðx i ÞPfX ¼ x i g: (2.230)

i¼ 1

If we consider that the random variable g { X } is a new variable Y , and y i ¼ g ( x i ),
then, according to ( 2.220 ), we have:
EfgðxÞg ¼ EfYg¼ X
i

y i PfY ¼ y i g: (2.231)

Generally, the number of values of the randomvariables X and Y may be different.
The expression in ( 2.231 ) is an alternative relation used to find the expected value of
the function of the variable X , g ( X ). Note that in ( 2.231 ), we have to find the values y i
and the probabilities P { Y ¼ y i }.
Example 2.7.5 Consider the random variable X with values x 1 ¼ 1, x 2 ¼ 1,
x 3 ¼ 2, x 4 ¼ 2. The corresponding probabilities are: P { x i } ¼ 1/4, for i ¼ 1,

,4.

...

Find the expected value of X 2 .

Solution Using ( 2.230 ), we have:

g¼ X

4

EfgðXÞg ¼ EfX 2 gðx i ÞPfX ¼ x i g;

i¼ 1

2

4 þ 1 2

2

4 þ 2 2 ¼ð 1 Þ 1 = 1 = 4 þð 2 Þ 1 = 1 = 4 ¼ 5 :

(2.232)

= 2

In order to use ( 2.231 ), we define the random variable Y ¼ X 2 , and find the
values x i 2 , i ¼ 1, ,4:

...

x 1 ¼ x 2 ¼ 1 x 3 ¼ x 4 ¼ 4 : (2.233)

;