Biomedical Engineering Reference
In-Depth Information
φ
f
(
α
)
x
(t)
1
1
t
0
.5
1
10
30
50
(a)
(b)
FIGURE 5.2:
(a) PDF and (b) characteristic function magnitude for continuous RV with uniform
distribution on [0
,
1].
and
2
b
a
)
2
12
1
b
+
a
(
b
−
2
x
σ
=
α
−
d
α
=
.
(5.9)
b
−
a
2
a
The characteristic function can be found as
exp
j
b
+
a
2
t
b
(
b
−
a
)
/
2
1
e
j
α
t
d
e
j
α
t
d
φ
x
(
t
)
=
α
=
α.
b
−
a
b
−
a
a
−
(
b
−
a
)
/
2
Simplifying with the aid of Euler's identity,
t
sin
b
−
a
t
exp
j
a
+
b
2
φ
x
(
t
)
=
.
(5.10)
b
−
a
2
t
2
Figure 5.2 illustrates the PDF and the magnitude of the characteristic function for a continuous
RV uniformly distributed on [0
,
1]. Note that the characteristic function in this case is not
=
φ
x
(
t
).
periodic but
φ
x
(
−
t
)
Drill Problem 5.1.1.
4
) is tossed once. Let x be a
random variable equaling ten times the number tossed. Determine: (a) p
x
(20), (b) P(
10
A pentahedral die (with faces labeled
0
,
1
,
2
,
3
,
≤
x
≤
50
),
2
(c) E(x), (d)
σ
x
.
Answers: 20, 0.8, 200, 0.2.
Drill Problem 5.1.2.
Random variable x is uniformly distributed on the interval
[
−
1
,
5]
. Deter-
2
mine: (a) F
x
(0), (b) F
x
(5)
,
(c)
η
x
, (d)
σ
x
.
Answers: 1
,
1
/
6
,
3
,
2.
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