Digital Signal Processing Reference
In-Depth Information
Answer
The random variable
X
is the discrete random variable and it has only two
values: 0 and 1, with the probabilities
Pf
0
g¼
5
=
8
;
Pf
1
g¼
3
=
8
:
(2.448)
The corresponding PDF has two delta functions in the discrete points 0 and 1, as
shown in Fig.
2.55a
,
f
X
ðxÞ¼Pf
0
gdðxÞþPf
1
gdðx
1
Þ¼
5
=
8
dðxÞþ
3
=
8
dðx
1
Þ:
(2.449)
The distribution function has steps at the discrete points 0, and 1, as shown in
Fig.
2.55b
.
F
X
ðxÞ¼Pf
0
guðxÞþPf
1
guðx
1
Þ¼
5
=
8
uðxÞþ
3
=
8
uðx
1
Þ:
(2.450)
Exercise E.2.2
Solve the previous problem but define the random variable
X
as the
random instant in time in which the call is received.
Answer
This example illustrates that one can define different variables for the
same random experiment. In this case, the random variable
X
is the continual
variable in the interval [0, 8].
If
x >
8 (the call has been received), then
F
X
ðxÞ¼PfX xg¼
1
:
(2.451)
For 0
x
8,
x
8
0
¼
x
8
:
F
X
ðxÞ¼PfX xg¼
(2.452)
For
x <
0 (the call cannot be received), resulting in:
F
X
ðxÞ¼PfX xg¼
0
:
(2.453)
Fig. 2.55
(
a
) PDF and (
b
) distribution in Exercise E.2.1
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