12_Sim_Hw3_Sol

# 12_Sim_Hw3_Sol - IEOR 4404 Assignment #3 Solutions...

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Unformatted text preview: IEOR 4404 Assignment #3 Solutions Simulation February 12, 2012 Prof. Mariana Olvera-Cravioto Page 1 of 3 Assignment #3 Solutions 1. var ( c 1 X 1 + c 2 X 2 ) = E [( c 1 X 1 + c 2 X 2 ) 2 ]- E [ c 1 X 1 + c 2 X 2 ] 2 = E [ c 2 1 X 2 1 + c 2 2 X 2 2 + 2 c 1 c 2 X 1 X 2 ]- ( c 1 E [ X 1 ] + c 2 E [ X 2 ]) 2 = c 2 1 E [ X 2 1 ] + c 2 2 E [ X 2 2 ] + 2 c 1 c 2 E [ X 1 X 2 ]- c 2 1 E [ X 1 ] 2- c 2 2 E [ X 2 ] 2- 2 c 1 c 2 E [ X 1 ] E [ X 2 ] = c 2 1 ( E [ X 2 1 ]- E [ X 1 ] 2 ) + c 2 2 ( E [ X 2 2 ]- E [ X 2 ] 2 + 2 c 1 c 2 ( E [ X 1 X 2 ]- E [ X 1 ] E [ X 2 ]) = c 2 1 var ( X 1 ) + c 2 2 var ( X 2 ) + 2 c 1 c 2 cov ( X 1 X 2 ) 2. (a) E [ X ( n )] = E [ n i =1 i X i ] = n i =1 i E [ X i ] = n i =1 i = n i =1 i = . Hence, X ( n ) is an unbiased estimator for . (b) Assume that the samples are drawn from an i.i.d distribution. Since X ( n ) is an unbiased estimator for , E [ ( X ( n )- ) 2 ] = V ar [ X ( n )] = V ar [...
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## This note was uploaded on 03/18/2012 for the course IEOR 4404 taught by Professor C during the Spring '10 term at Columbia.

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12_Sim_Hw3_Sol - IEOR 4404 Assignment #3 Solutions...

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