Digital Signal Processing Reference
In-Depth Information
do not ensure, for example, that events {
A
1
A
2
} and {
A
3
} are independent.
The former is ensured by adding
PfA
1
A
2
A
3
g¼PfA
1
gPfA
2
gPfA
3
g¼PfA
1
; A
2
gPfA
3
g
¼ P ðA
1
; A
2
Þ; A
3
f
g:
(1.135)
A.1.13. It can, if the events are equivalent,
A ¼ B
. Then,
ðA þ BÞ¼ðA [ BÞ¼A ¼ B;
ðABÞ¼ðA \ BÞ¼A ¼ B:
(1.136)
For example, [WEN86, p. 33], if, in a communication system, a message
can only be distorted by a noise, occupying the same time interval, then,
A ¼f
''the message is distorted''
g:
(1.137)
B ¼f
''there is a noise occupying the same time interval as the message''
g;
(1.138)
are equivalent and (
1.136
) holds.
A.1.14. Bayes's rule is useful for calculating conditional probabilities
P
{
A
|
B
},
since in many problems it may be difficult to directly calculate
P
{
A
|
B
}.
However, calculating
P
{
B
|
A
} may be straightforward [MIL04, p. 23].
Additionally, in some cases
P
{
B
|
A
} may be known as a characteristic of
the system.
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