Assignment07Sol

# Assignment07Sol - Columbia University IEOR 4404 Simulation...

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ) ....
View Full Document

## 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.

### Page1 / 13

Assignment07Sol - Columbia University IEOR 4404 Simulation...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online