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10SimHw3Solution

# 10SimHw3Solution - IEOR E4404.001 SIMULATION Prof Mariana...

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IEOR E4404.001 SIMULATION Prof. Mariana Olvera-Cravioto Assignment #3 Solutions 1. The following MATLAB code computes 95% approximate conﬁdence intervals for the expected number dice rolls that are needed: N = 1000; counts = []; for n=1:N outcomes = zeros(1,11); count = 0; while( prod( outcomes ) == 0 ) % generate a dice roll dice1 = floor( 6 * rand() ) + 1; dice2 = floor( 6 * rand() ) + 1; total = dice1 + dice2; outcomes( total - 1 ) = 1; count = count + 1; end counts = [ counts count ]; end m = mean( counts ); sigma = std( counts ); CI = [ m - 1.96 * sigma/N^.5, m + 1.96 * sigma/N^.5 ] The vector outcomes is used to keep track of the 11 possible outcomes. It is initialized as zeros(1,11) and then the ( i - 1)th component of outcomes is set to 1 if outcome i is rolled. The while loop iterates until all components of outcomes are nonzero. Running the code gives the following output: CI = 60.7802 62.1818

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k = 10 3 k = 10 4 k = 10 5 exact Cov ( U,e U ) [0 . 1017; 0 . 1961] [0 . 1315; 0 . 1618] [0 . 1376; 0 . 1471] 0 . 1409 Table 1: For Question 2 2. The exact answer is: Cov (U ,e U ) = E [ Ue U ] - E [ U ] E [ e U ] = 1 0 xe x dx - 1 2 ( e - 1) = [ xe x ] 1 0 - 1 0 e x dx - 1 2 ( e - 1) = 1 - 1 2 ( e - 1) 0 . 1409 On the other hand, the Monte Carlo approximation of Cov ( U,e U ) is simply given by: 1 k k i =1 U i e U i - ( 1 k k i =1 U i ) ( 1 k k i =1 e U i ) We provide with a 95% Conﬁdence Interval of Monte Carlo approximation of Cov ( U,e U ). 3. According to the acceptance-rejection algorithm, the probability of rejection is
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10SimHw3Solution - IEOR E4404.001 SIMULATION Prof Mariana...

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