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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
3.30. Given the joint probability density function
f
XY
(x, y) =
x(1 + 3y
2
)/4, 0 < x < 2, 0 < y < 1
0, elsewhere,
derive the marginal densities f
X
(x), f
Y
(y), the conditional density f
X|Y
(x|y) and evaluate the prob-
ability
Pr
1
4
< X <
1
2
Y =
1
3
.
Solution: The marginal densities are
f
X
(x) =
1
0
f
XY
(x, y)dy =
1
0
1
4
x(1 + 3y
2
)dy
=
1
4
x
y + y
3
1
0
=
x
2
, 0 < x < 2
f
Y
(y) =
2
0
f
XY
(x, y)dx =
2
0
1
4
x(1 + 3y
2
)dx
=
(1 + 3y
2
)
4
x
2
2
2
0
=
1 + 3y
2
2
, 0 < y < 1
The conditional density is
f
X|Y
(x|y) =
f
XY
(x, y)
f
Y
(y)
=
x(1 + 3y
2
)/4
(1 + 3y
2
) /2
=
x
2
, 0 < x < 2
Then, the probability is
Pr
1
4
< X <
1
2
Y =
1
3
=
1/2
1/4
f
X|Y
(x|y = 1/3)dx
=
x
2
4
1/2
1/4
=
1
4
1
4
1
16
=
3
64
61
3.31. Find the marginal densit ...
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Publisher Resources

ISBN: 9781439849897