Digital Signal Processing Reference
In-Depth Information
1.5 Conditional Probability
Often the occurrence of one event may be dependent upon the occurrence of
another. If we have two events
A
and
B
, we may wish to know the probability of
A
if we already know that
B
has occurred. Given an event
B
with nonzero
probability,
PfBg >
0
;
(1.42)
we define such probability as conditional probability
P
{
A
|
B
},
P
f
A
\
B
g
PfBg
P
f
AB
g
PfBg
;
PfAjBg¼
¼
(1.43)
where
P
{
AB
} is the probability of the joint occurrence of
A
and
B
. It is called a
joint
probability
.
Equation (
1.43
) can be interpreted, depending on the relations between
A
and
B
,
as described in the following.
The event
A
occurs, if and only if, the outcome
s
i
is in the intersection of events
A
and
B
(i.e., in
A\B
) as shown in Fig.
1.7a
.
If we have
A ¼ B
(Fig.
1.7b
), then, given that event
B
has occurred, event
A
is a
certain event with the probability 1. The same is confirmed from (
1.43
), making
PfABg¼PfBg:
(1.44)
The other extreme case is when events
A
and
B
do not have elements in common
(Fig.
1.7c
), resulting in the conditional probability
P
{
A
|
B
} being equal to zero.
The conditional probability, since it is still probability, must satisfy the three
axioms (
1.11
)-(
1.13
).
Axiom I
The first axiom is obviously satisfied, because
P
{
B
} is positive, and the
joint probability
PfABg¼PfA \ Bg¼PfAgþPfBgPfA [ Bg
(1.45)
is also positive because the following relation holds:
PfA þ Bg¼PfA [ Bg¼PfAgþPfBgPfA \ BgPfAgþPfBg:
(1.46)
Axiom II
The second axiom is satisfied because
P
f
SB
g
PfBg
¼
PfBg
PfBg
¼
1
PfSjBg¼
:
(1.47)
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