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4703-10-Fall-h7

# 4703-10-Fall-h7 - IEOR 4703 Homework 7 1 Consider the...

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IEOR 4703: Homework 7 1. Consider the problem of estimating θ ( 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 θ ( x ) (1 - Φ( x )) e x where Φ( . ) is the CDF of a standard normal random variable. (b) Show that E [ X 2 ] (1 - Φ( 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 θ ( 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 θ ( x ) e 1 - ( x - 1) 2 / 2 . (In particular θ ( 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 L’Hopitals rule for computing the limit.) This means that the improvement factor of the importance sampling algorithm over the standard simulation algorithm tends to infinity as the event becomes rarer.

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4703-10-Fall-h7 - IEOR 4703 Homework 7 1 Consider the...

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