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problem2_12 - U U J U U 2 1 2 2 1 2 = J e e r r =-=-1 2 1 2...

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SAT 6934 MONTE CARLO SATISTICAL METHODS Problem #2.12 Sato Klein (a)Box-Muller Algorithm U 1 and U 2 are iid U(1,0) X U U X U U 1 1 2 2 1 2 2 2 2 2 = - = - log( )cos( ) log( )sin( ) π π The inverse transformations are U X X U X X 1 1 2 2 2 2 1 1 2 1 2 1 2 = - + = - exp{ ( )} tan ( ) π Joint density of(X 1 ,X 2 )is f X1,X2 (X 1 ,X 2 )=f U1,U2 ( , ) U U J 1 2 J X X X X X X X X X X X X X = - + - - + + - + - 1 1 2 2 2 2 1 2 2 2 2 1 2 2 1 2 2 1 2 2 1 2 1 2 1 2 1 1 2 1 1 exp{ ( )} exp{ ( )} ( ) ( ) π π = - + 1 2 1 2 1 2 2 2 π exp{ ( )} X X f X1,X2 (X 1 ,X 2 )=f U1,U2 (exp{ ( ), tan ( )) - + - 1 2 1 2 1 2 2 2 1 1 2 X X X X J π = - + 1 2 1 2 1 2 2 2 π exp{ ( )} X X = - - 1 2 1 2 1 2 2 2 2 2 π π e e X X Hence,X 1 and X 2 are iid N(0,1) (b)Set r 2 =-2logU 1 , θ =2 π U 2 The inverse transformations are U r U 1 2 2 2 = - = exp( ) θ π Joint density of(r
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Unformatted text preview: U U J U U 2 1 2 2 1 2 , ( , ) ( , ) = J e e r r =-=--1 2 1 2 1 2 1 2 2 2 2 2 fr r f U U J e U U r 2 1 2 2 2 1 2 2 1 2 1 2 , ( , ) ( , ) = =-Hence, r 2 is distributed chi square and θ is distributed uniform(0,2 π ). (c) Establish –2logU=r 2 since r 2 ~ 2 2 = exp(2) and –2logU ~ exp(2)...
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