Digital Signal Processing Reference
In-Depth Information
This property is easy to justify knowing that according to (
2.10
) the distribution
is the probability and thus must have the values between 0 and 1.
F
X
ð1Þ ¼
0
;
P.2
F
X
ð1Þ ¼
1
:
(2.42)
The first equation follows from the fact that
Pfx 1g
is the probability of the
empty set and hence equal to 0. The second equation follows from the probability of
a certain event which in turns is equal to 1.
F
X
ðx
1
ÞF
X
ðx
2
Þ
for
x
1
< x
2
:
(2.43)
P.3
This property states that distribution is a nondecreasing function. For
x
1
< x
2
,
the event
X x
1
is included in the event
X x
2
and consequently
PfX x
1
g
PfX x
2
g
.
P.4
F
X
ðx
þ
Þ¼F
X
ðxÞ;
(2.44)
where the denotation
x
+
0, which is infinitesimally small as
e !
0. This property states that distribution is a
continuous from the right
function.
As we approach
x
from the right, the limiting value of the distribution should be the
value of the distribution in this point, i.e.,
F
X
(
x
).
The properties (
2.41
)-(
2.44
), altogether, may be used to test if a given function
could be a valid distribution function.
Next, we relate the probability that the variable
X
is in a given interval, with its
distribution function,
implies
x
+
e
,
e >
Pfx
1
<X x
2
g¼PfX x
2
gPfX x
1
g:
(2.45)
Using the definition (
2.10
) from (
2.45
), it follows:
Pfx
1
<X x
2
g¼F
X
ðx
2
ÞF
X
ðx
1
Þ:
(2.46)
Example 2.2.5
Using (
2.45
), find the following probabilities for the random variable
X
from Example 2.2.1:
(a)
Pf
1
:
1
<X
1
:
8
g:
(2.47)
(b)
Pf
0
:
5
<X
1
:
5
g:
(2.48)
(c)
Pf
0
:
4
<X
1
g:
(2.49)
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