Digital Signal Processing Reference
In-Depth Information
Fig. 1.7
Events
A
and
B
Axiom III
To prove Axiom III, consider the union of event
A
with a mutually
exclusive event
C
in the conditional probability,
P
f
A
[
C
;
B
g
PfBg
P
f
A
\
B
;
C
\
B
g
PfBg
P
fð
A
\
B
Þ[ð
C
\
B
Þg
PfBg
PfA [ CjBg¼
¼
¼
P
f
A
\
B
gþ
P
f
C
\
B
g
P
fð
A
\
B
Þ\ð
C
\
B
Þg
PfBg
¼
:
(1.48)
Since
A
and
C
are mutually exclusive, then events (
A\B
) and (
C\B
) are also
mutually exclusive, as shown in Fig.
1.8
, resulting in:
PfðA \ BÞ\ðC \ BÞg ¼
0
:
(1.49)
Placing (
1.49
) into (
1.48
) and using the definition of conditional probability
(
1.43
), we easily show that the Axiom III holds:
P
f
A
\
B
gþ
P
f
C
\
B
g
PfBg
P
f
A
\
B
g
PfBg
P
f
C
\
B
g
PfBg
PfA [ CjBg¼
¼
þ
¼ PfAjBgþPfCjBg
(1.50)
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