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08SimHw3sol - IEOR 4404 Simulation Prof Mariana...

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IEOR 4404 Assignment #3 Solutions Simulation October 15, 2008 Prof. Mariana Olvera-Cravioto Page 1 of 4 Assignment #3 Solutions 1. x 1 = 10, x 2 = 20, x 3 = 40, x 4 = 80, x 5 = 10, x 6 = 20, x 7 = 40, x 8 = 80, x 9 = 10, x 10 = 20. 2. (a) Using the substitution u = ( x + 2) / 4 or x = 4 u - 2, we get Z 2 - 2 e x + x 2 dx = Z 1 0 4 e 16 u 2 - 12 u +2 du Using simulation, where h ( u ) = 4 e 16 u 2 - 12 u +2 , Z 1 0 4 e 16 u 2 - 12 u +2 du 1 n n X i =1 h ( U i ) 93 . 0677 where n = 10 6 . (b) Integrating gives Z 0 x (1 + x 2 ) - 2 dx = 1 2 Z 1 1 u 2 du = - 1 2 u fl fl fl fl 1 = 1 2 Using the substitution u = (1 + x 2 ) - 1 / 2 , du = - x (1 + x 2 ) - 3 / 2 dx we get Z 0 x (1 + x 2 ) - 2 dx = - Z 0 1 u du = Z 1 0 u du Using simulation with h ( u ) = u , Z 0 x (1 + x 2 ) - 2 dx 1 n n X i =1 U i 0 . 5001 where the U i are iid uniform (0,1) random variables, n = 10 6 . (c) Z 1 0 Z 1 0 g ( x, y ) dxdy = Z 1 0 Z 1 0 e ( x + y ) 2 dxdy 1 n n X i g ( X i , Y i ) where X i and Y i are independent Uniform(0,1) random variables and g ( x, y ) = e ( x + y ) 2 . Using simulation, R 1 0 R 1 0 e ( x + y ) 2 dxdy 4 . 9017, when n = 10 7 . 3. We can use the following MATLAB code: m = 100; % Number of simulation runs mSum = 0; % Initialize cumulative sum
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