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Probabilistic Models for Dynamical Systems, 2nd Edition
book

Probabilistic Models for Dynamical Systems, 2nd Edition

by Haym Benaroya, Seon Mi Han, Mark Nagurka
May 2013
Intermediate to advanced content levelIntermediate to advanced
764 pages
6h 43m
English
CRC Press
Content preview from Probabilistic Models for Dynamical Systems, 2nd Edition
The marginal density is obtained by integrating f
W Z
(w, z) over the shaded region, D. For convenience, let
f
1
= y = z w
f
2
= 2 y = 2 z + w.
Then,
f
Z
(z) =
D
f
W Z
(w, z) dw
=
z
0
f
1
dw 0 z < 1
1
z1
f
1
dw +
z1
0
f
2
dw 1 z < 2
1
z2
f
2
dw 2 z < 3
That is,
f
Z
(z) =
z
2
/2, 0 z < 1
z
2
+ 3z 3/2 1 z < 2
z
2
/2 3z + 9/2 2 z < 3
0 elsewhere
.
106
4.17. List the sequence of transformations needed to derive the general probability density function f
Y
5
for
Y
5
=
X
2
1
+ X
2
2
+ X
3
X
4
,
assuming that all f
X
i
are known.
Solution: The following transformations are needed:
1. X
1
−→ X
2
1
2. −→ X
2
1
+ X
2
3. −→
X
2
1
+ X
2
2
4. −→ X
3
X
4
5. −→
X
2
1
+ X
2
2
+ X
3
X
4
.
107
4.18. List the sequence
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Publisher Resources

ISBN: 9781439849897