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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
12.5. Use the composition method to generate random numbers distributed according to
f
Y
(y) =
1
4
sin y +
9
32π
3
y
2
, 0 < y 2π.
Solution: Let us write
f
Y
(y) =
1
4
f
I
(y) +
3
4
f
II
(y) ,
where
f
I
(y) =
1
4
sin
y
2
and f
II
(y) =
3
8π
3
y
2
F
I
(y) =
1
2
cos
y
2
and F
II
(y) =
1
8π
3
y
3
.
The random numbers y are generated by
y = F
1
I
(x
2
) = 2 cos
1
(2x
2
) , if 0 x
1
<
1
4
y = F
1
II
(x
2
) = 2π
3
x
2
, if
1
4
x
1
<
3
4
.
where x
1
and x
2
are uniform random numbers between 0 and 1. The Table below illustrates this with
numerical examples.
x
1
0.23 0.89 0.019 0.49 0.76 0.60 0.97 ···
x
2
0.44 0.92 0.20 0.61 0.74 0.94 0.18 ···
y 5.29 6.11 3.96 5.33 5.68 6.16 3.55 ···
317
12.6. Use Von Neumann’s rejection-acceptance method
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Publisher Resources

ISBN: 9781439849897