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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
4 Functions of Random Variables
Section 4.1: Exact Functions of One Variable
4.1. Derive the probability density function for Y, given that Y = X
4
, for the following densities. Sketch
all density functions.
(a) f
X
(x) =
1
2
, 0 < x < 2
(b) f
X
(x) = c exp(x), 0 < x < .
Solution: For any density function we need to use the following:
g(x) = X
4
g
(x) = 4X
3
.
Mathematically there are four roots (obtained by defining Y = Z
2
and Z = X
2
and performing successive
square roots) given by
Y = Z
2
Z
1,2
= ±
Y
Z = X
2
X
1,2
= ±
Z = ±
*
±
Y .
For a particular realization y then, the four roots are
x
1
=
4
y, x
2
=
4
y,
x
3
= i
4
y, x
4
= i
4
y.
In general then,
f
Y
(y) =
4
i=1
f
X
(x
i
)
|g
(x
i
)|
.
(a)
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Publisher Resources

ISBN: 9781439849897