Digital Signal Processing Reference
In-Depth Information
The mean value also never occurs as a value of the random variable, as shown in
Fig.
2.39
.
2.7.2.1 General Expressions
More general expression of (
2.220
) can be given as:
EfXg¼
1
i¼1
x
i
PfX ¼x
i
g:
(2.223)
However, the infinite sum (
2.223
) may not converge and in that case, a mean
value does not exist. The condition for the existence of a mean value is the absolute
convergence of the sum (
2.223
), as given in (
2.224
):
1
xj PfX ¼ x
i
g<1:
(2.224)
i¼1
Example 2.7.4
Consider a randomvariable
X
with the values
x
k
¼ k
,
k ¼
0,1,
,
1
.
...
The probability that random variable takes the value
k
is given as:
=p
2
k
2
4
PfX ¼ kg¼
;
(2.225)
where
1
=p
2
k
2
4
¼
1
:
(2.226)
k¼
0
The mean value of the random variable
X
is:
EfXg¼
1
k¼
0
kPfX ¼ kg¼
1
k¼
0
p
2
1
k¼
0
=p
2
k
2
4
4
1
k
:
k
¼
(2.227)
Fig. 2.39
Illustration of Example 2.7.3
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