AMDA-2008-Session 2

# AMDA-2008-Session 2 - -U u u F x x π Accept-Reject Method...

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Generation of Random Variates

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Probability Integral Transform F. on distributi the has ) ( F variable random then the Unif(0,1) ~ U If } ) ( : inf{ ) ( F Let cdf. a is F Suppose - - u u x F x u =
Exponential Distribution ) Exp( as d distribute is lnU - X variable random then the ,1) Uniform(0 as d distribute is U If λ λ =

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Gamma Distribution freedom. of degrees 2n on with distributi square - chi the called is on distributi then the 2 If . and n parameters on with distributi Gamma a follows ) ln(U - thenY variables random ,1) Uniform(0 d distribute y identicall t independen are U ,..., U , U If n 1 j j n 2 1 = = = β β β
Beta Distribution on. distributi ,1) Uniform(0 get the then we 1 If . and m parameters with on distributi Beta a follows ) ln( ) ln( thenY variables random ,1) Uniform(0 d distribute y identicall t independen are U ,..., U , U If 1 1 n m 2 1 = = = + = = + n m n U U n m j j m j j

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Normal Distribution (Box-Muller Algorithm) N(0,1) from draws t independen are x and x . 3 ) 2 sin( ) ln( 2 x ) 2 cos( ) ln( 2 x Define 2. ,1) Uniform(0 iid , U Generate . 1 2 1 2 1 2 2 1 1 2 1 U U U U U π π - = - =
Cauchy Distribution ) Cauchy(0,1 is 2 1 tan then Unif(0,1) ~ U if Then . 2 1 tan ) ( 2 tan 1 F(x) is cdf The x all for ) 1 ( 1 f(x) by given is on distributi ) Cauchy(0,1 the of pdf The 1 2 - - =

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Unformatted text preview: --U u u F x x π Accept-Reject Method f. to according d distribute is Y variate The otherwise 1 Return to . 3 Mg(X) f(X) U if X Y Accept 2. Unif(0,1) ~ U g, ~ X Generate 1. constant. a is M where x, all for Mg(x) f(x) such that pdfs be g and f Let ≤ = ≤ Example N(0,1) from generate to method reject -accept apply the can Then we . e 2/ M Hence 1. at x attained is which , by bounded is ) 1 ( g(x) f(x) Now, on. distributi N(0,1) from generate want to We ons. distributi ) Cauchy(0,1 and N(0,1) of pdfs be g and f Suppose 2 2 2 = ± = + =-x e x Discrete Distributions ) p (Assume k. X put then p If Unif(0,1). ~ U Generate . be X variable random the of on distributi y probabilit Let the 1-k 2 1 2 1 = = ≤ < k n n p U p p p x x x...
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