Digital Signal Processing Reference
In-Depth Information
Finally, from (
3.357
) and (
3.363
) we arrive at:
g
1
ðx
1
Þ¼f
X
1
ðx
1
Þ;
g
2
ðx
2
Þ¼f
X
2
ðx
2
Þ;
(3.364)
which confirms that the condition is also a sufficient condition for the
independence.
A.3.6. Yes. It is true, as explained in the following:
The event
A ¼fa < X
1
b; c < X
2
dg:
(3.365)
can be presented as a difference between the events
A
1
and
A
2
, as shown in
Fig.
3.38
.
A
1
¼fa < X
1
b; X
2
dg; A
2
¼fa < X
1
b; X
2
cg;
(3.366)
PfAg¼PfA
1
gPfA
2
g:
(3.367)
Placing (
3.365
) and (
3.366
) into (
3.367
), we get:
Pfa < X
1
b; c < X
2
dg¼Pfa < X
1
b; X
2
dgPfa < X
1
b; X
2
cg:
(3.368)
Fig. 3.38
Illustration of A.3.6
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