Assignment07Sol - Columbia University IEOR 4404 Simulation...

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Unformatted text preview: Columbia University IEOR 4404: Simulation Spring 2011 Solution to Assignment 7 1. Solution: (a) u1D438 [ u1D43C ] = u1D443 [ u1D44C < u1D454 ( u1D44B )] = u1D443 [ u1D44Fu1D448 2 < u1D454 ( u1D448 1 )] = uni222B.alt02 1 u1D443 [ u1D44Fu1D448 2 < u1D454 ( u1D448 1 ) ∣ u1D448 1 = u1D465 ] u1D451u1D465 = uni222B.alt02 1 u1D443 [ u1D44Fu1D448 2 < u1D454 ( u1D465 )] u1D451u1D465 = uni222B.alt02 1 u1D443 uni005B.alt03 u1D448 2 < u1D454 ( u1D465 ) u1D44F uni005D.alt03 u1D451u1D465 = uni222B.alt02 1 u1D454 ( u1D465 ) u1D44F u1D451u1D465 = 1 u1D44F uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 (b) First, we compute Var( u1D44Fu1D43C ): Var( u1D44Fu1D43C ) = u1D44F 2 Var( u1D43C ) = u1D44F 2 uni0028.alt01 u1D438 [ u1D43C 2 ] − u1D438 [ u1D43C ] 2 uni0029.alt01 = u1D44F 2 uni0028.alt01 u1D438 [ u1D43C ] − u1D438 [ u1D43C ] 2 uni0029.alt01 = u1D44F 2 uni005B.alt04 1 u1D44F uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 − 1 u1D44F 2 uni0028.alt03uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 uni0029.alt03 2 uni005D.alt04 = u1D44F uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 − uni0028.alt03uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 uni0029.alt03 2 Next, we compute Var( u1D454 ( u1D448 )): Var( u1D454 ( u1D448 )) = u1D438 [ u1D454 ( u1D448 ) 2 ] − u1D438 [ u1D454 ( u1D448 )] 2 = uni222B.alt02 1 u1D454 ( u1D465 ) 2 u1D451u1D465 − uni0028.alt03uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 uni0029.alt03 2 Thus, since u1D454 ( u1D465 ) ≤ u1D44F for 0 ≤ u1D465 ≤ 1, Var( u1D44Fu1D43C ) = u1D44F uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 − uni0028.alt03uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 uni0029.alt03 2 ≥ uni222B.alt02 1 u1D454 ( u1D465 ) 2 u1D451u1D465 − uni0028.alt03uni222B.alt02 1 u1D454 ( u1D465 ) u1D451u1D465 uni0029.alt03 2 = Var ( u1D454 ( u1D448 )) . (1) 2. (a) Repeat u1D45B simulation runs. In run u1D456 , for 1 ≤ u1D456 ≤ u1D45B , use independent random numbers u1D448 1 ,u1D456 and u1D448 2 ,u1D456 to generate independent binomial ( u1D45B,u1D45D ) random variables u1D44B u1D456 and u1D44C u1D456 using the inverse transform algorithm (see page 57 of the textbook). Estimate u1D703 by ˆ u1D703 ≡ 1 u1D45B u1D45B uni2211.alt02 u1D456 =1 u1D452 u1D44B u1D456 u1D44C u1D456 (b) u1D44Bu1D44C can be used as a control variate. In order to show that using u1D44Bu1D44C as a control variate gives us an estimator with smaller variance, it suffices to show that u1D44Bu1D44C is correlated to u1D452 u1D44Bu1D44C . But this follows from the fact that u1D452 u1D44Bu1D44C is a strictly increasing function of u1D44Bu1D44C (see page 140). Since u1D438 [ u1D44Bu1D44C ] = u1D45B 2 u1D45D 2 , our estimator for u1D703 is u1D452 u1D44Bu1D44C + u1D450 ( u1D44Bu1D44C − u1D45B 2 u1D45D 2 ) ....
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This note was uploaded on 01/16/2012 for the course IEOR 4404 taught by Professor Joseblanchet during the Spring '11 term at Columbia College.

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Assignment07Sol - Columbia University IEOR 4404 Simulation...

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