Random numbers from an arbitrary distribution SSP 5

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Lecture 8

• Random numbers from an arbitrary distribution
  (SSP 5)

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Distribution xx1 p(x)p1 xx2 p(x)p2 (1-p1)
Call ? if ? lt p1 set xx1 else xx2
Distribution xx1 p(x)p1 xx2 p(x)p2 xx3 p
(x)p3
Call ? if ? lt p1 set xx1 else if ?ltp1p2
xx2 else xx3
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Sampling from a discrete distribution
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Sampling from a continuous distribution, p(x)dx,
a?x?b
Method of inversion Solve P(x)?, where
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Example the exponential distribution
or x -? ln?
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Example The power law distribution
xx0?-1/(?-1)
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Method of Composition
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Von Neumanns method
To select from p(x)dx, a?x?b
Write p(x)dxC Q(x) ?(x)dx, Where maxQ(x)1
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Von Neumanns Acceptance-Rejection algorithm
?(x) the sampling, or targeting function, must
be a distribution which can be easily
sampled, should mimic p(x) as far as possible so
acceptance is high Q(x) the acceptance function
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Good and not so good sampling functions
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Method of Superposition
Write p(x)dx in the form,
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Example Angular distribution in Thompson
scattering
is a probability distribution,
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Show that the algorithm is If ?1 lt ¾ set x2 ?2-1
else set x?(2 ?2 1)1/3, the sign chosen at
random.