Digital Signal Processing Reference
In-Depth Information
Similarly, for
x > b
, the probability is equal to:
PfðX xÞ\ða<X bÞg¼Pfða<X bÞ\ða<X bÞg¼Pfa<X bg;
(2.125)
resulting in:
P
f
a
<
X
b
g
Pfa<X bg
¼
1
F
X
ðxja<X bÞ¼
:
(2.126)
Finally, from (
2.122
), (
2.124
), and (
2.126
), we have:
8
<
1
for
x >b;
F
X
ð
x
Þ
F
X
ð
a
Þ
F
X
ðbÞF
X
ðaÞ
F
X
ðxja<X bÞ¼
for
a< x b
;
(2.127)
:
0
for
x a:
Using (
2.106
), from (
2.127
), we obtain the corresponding density function as:
8
<
0
for
x >b;
f
X
ð
x
Þ
F
X
ð
a
Þ
F
X
ðbÞF
X
ðaÞ
f
X
ðxja<X bÞ¼
for
a< x b;
(2.128)
:
0
for
x a:
Example 2.5.2
Consider the uniform random variable
X
in the interval [0, 4]. Let the
conditional event
B
be defined as
Pf
0
<X
1
g
(i.e.,
a ¼
0 and
b ¼
1in(
2.118
)).
From (
2.128
), we get:
8
<
0
for
x>
1
;
f
X
ðxÞ
F
X
ð
1
ÞF
X
ð
0
Þ
¼
1
=
4
f
X
ðxj
0
<X
1
Þ¼
4
0
¼
1
for
0
< x
1
;
(2.129)
:
1
=
0
for
x
0
:
The PDF of the random variable
X
(solid line) and the conditional PDF (
2.129
)
(dashdot line) are shown in Fig.
2.27a
.
From (
2.127
), the conditional distribution is:
8
<
1
for
x>
1
;
ð
1
=
4
Þx
F
X
ðxj
0
<X
1
Þ¼
4
0
¼ x
for
0
< x
1
;
(2.130)
:
1
=
0
for
x
0
:
The conditional distribution (dashdot line), along with the distribution of the
random variable
X
, are shown in Figs.
2.27a
,
b
. Observe that the conditioned
distribution has all of the characteristics of the distribution itself.
Search WWH ::
Custom Search