Digital Signal Processing Reference
In-Depth Information
where
1
=
2
for
0
< x
2
<
2
f
X
1
ðx
1
Þ¼
:
(3.68)
0
otherwise
1
for
0
< x
2
<
1
f
X
2
ðx
2
Þ¼
:
(3.69)
0
otherwise
From (
3.65
) and (
3.67
)-(
3.69
), we can conclude that
the variables are
independent.
The result (
3.65
) can be generalized to
N
jointly independent random variables:
F
X
1
;
...
;X
N
ðx
1
;
...
; x
N
Þ¼
Y
N
F
X
i
ðx
i
Þ:
(3.70)
i¼
1
f
X
1
;
...
;X
N
ðx
1
;
...
; x
N
Þ¼
Y
N
f
X
i
ðx
i
Þ:
(3.71)
i¼
1
3.3 Expected Values and Moments
3.3.1 Expected Value
In order to find the mean value of two joint random variables
X
1
and
X
2
, we will
apply the similar procedure to that which we used in the case of one random
variable (see Sect.
2.7
), starting with a random experiment.
Consider two discrete random variables
X
1
and
X
2
with the possible values
x
1
i
and
x
2
j
, respectively.
The experiment is performed
N
times under the same conditions,
N ¼
X
X
N
1
N
2
N
ij
;
(3.72)
i¼
1
j¼
1
and as a result the following values are obtained:
X
1
¼ x
11
;
and
X
2
¼ x
21
;
N
11
times
:
X
1
¼ x
1
i;
and
X
2
¼ x
2
j
;
N
ij
times
:
(3.73)
X
1
¼ x
1
N
1
;
and
X
2
¼ x
2
N
2
;
N
N
1
N
2
times
:
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