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4703-10-Fall-h7-solu

# 4703-10-Fall-h7-solu - IEOR 4703 Solutions to Homework 7 1...

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IEOR 4703: Solutions to 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. SOLUTION: Observe that e z is an increasing function and e Z 1 { Z x } e x 1 { Z x } Hence, θ ( x ) = E [ e Z 1 { Z x } ] e x E [ 1 { Z x } ] = e x P ( Z x ) = (1 - Φ( x )) e x (b) Show that E [ X 2 ] (1 - Φ( x )) e 2 x SOLUTION: E [ X 2 ] = E [ e 2 Z 1 { Z x } ] e 2 x E [ 1 { Z x } ] = e 2 x P ( Z x ) = (1 - Φ( x )) e 2 x (c) Consider doing importance sampling with alternative density g ( y ) 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 } ] . SOLUTION: Let h ( y ) = e - y I { y > x } . Letting f ( y ) denote the N (0 , 1) density and g ( y ) the N ( x, 1) density, it is easily verified that the likelihood ratio L ( y ) = f ( y ) /g ( y ) is given by L ( y ) = e - y 2 / 2 e - ( y - x ) 2 / 2 = e x 2 / 2 - yx , and thus h ( W ) L ( W ) = e x 2 / 2 e W e - Wx I { W>x } = Y. From the general theory of importance sampling, we know that E ( X ) = E ( h ( W ) L ( W )) = E ( Y ) . Since e - Wx < 0 when W > x , we have Y e x 2 / 2 e W I { W>x } yielding E [ Y ] e - x 2 / 2 E [ e W I { W>x } ] . 1

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(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 } ].) SOLUTION: As e z + x e z + x for z 0 and x 1, E [ e W 1 { W>x } ] = E [ e Z + x 1 { Z> 0 } ] = Z 0 e z + x 1 2 π e - z 2 / 2 dz Z 0 e z + x 1 2 π e - z 2 / 2 dz = e x +1 / 2 Z 0 1 2 π e - ( z - 1) 2 / 2 dz e x +1 / 2 Hence θ ( x ) = E [ Y ] e - x
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4703-10-Fall-h7-solu - IEOR 4703 Solutions to Homework 7 1...

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