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Unformatted text preview: IEOR 4703: Homework 7 1. Consider the problem of estimating ( x ) = E [ e Z I { Z x } ] for x 1 and where Z N(0 , 1). Let X := e Z I { Z x } . (a) Show that ( x ) (1 ( x )) e x where ( . ) is the CDF of a standard normal random variable. (b) Show that E [ X 2 ] (1 ( x )) e 2 x (c) Consider doing importance sampling with alternative density g ( x ) having the density of W N( x, 1). (That is, we shift the mean by x .) Show that ( x ) = E [ Y ] where Y = e x 2 / 2 e W e Wx I { W>x } . Further show that E [ Y ] e x 2 / 2 E [ e W I { W>x } ] . (d) Using the fact that W may be expressed as Z + x , show that ( x ) e 1 ( x 1) 2 / 2 . (In particular ( x ) 0 as x , which can also be proved directly using the monotone convergence theorem on E [ e Z I { Z x } ].) (e) We will now develop an upper bound for the second moment using importance sampling. Show that E [ Y 2 ] e ( x 1) 2 / 2 e 3 . (f) Using the above results, show that var ( X ) var ( Y ) as x . (You should use LHopitals rule for computing the limit.) This means that the improvement factor of the importance sampling algorithm over...
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This note was uploaded on 03/14/2012 for the course IEOR 4703 taught by Professor Sigman during the Spring '07 term at Columbia.
 Spring '07
 sigman

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