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Unformatted text preview: IEOR 4404 Simulation Lecture 4 January 30th, 2012 Mariana OlveraCravioto [email protected] 1 Objective of simulation We use simulation to estimate quantities, such as probabilities, expectations, variances, etc., that are otherwise difficult (if not impossible) to calculate. We say that ˆ α is an unbiased estimator of a α if E [ˆ α ] = α It is often the case that we can find several unbiased estimators for α , and we will often prefer the estimators that have smaller variance. It is not always possible to find unbiased estimators for the quantities we want to analyze. There exist “biased” estimators that may be more appropriate than “unbiased” ones for certain problems. IEOR 4404, Lecture 4 2 Estimating the mean and the variance Suppose that X 1 ,X 2 ,...,X n are iid random variables with finite mean μ and finite variance σ 2 , both unknown. An unbiased estimator of μ is X ( n ) = 1 n n X i =1 X i and it is called the sample mean . An unbiased estimator of σ 2 is S 2 ( n ) = 1 n 1 n X i =1 ( X i X ( n )) 2 and it is called the sample variance . Note: p S 2 ( n ) is not an unbiased estimator of σ . IEOR 4404, Lecture 4 3 To see that X ( n ) is an unbiased estimator of μ = E [ X 1 ]: E [ X ( n )] = 1 n E " n X i =1 X i # = 1 n n X i =1 E [ X i ] = E [ X 1 ] To see that S 2 ( n ) is an unbiased estimator of σ 2 = Var( X 1 ): E [ S 2 ( n )] = 1 n 1 E " n X i =1 ( X i X ( n )) 2 # = 1 n 1 n X i =1 E [( X i X ( n )) 2 ] = 1 n 1 n X i =1 E [ X 2 i ] 2 E [ X i X ( n )] + E [ X ( n ) 2 ] = nE [ X 2 1 ] n 1 2 n ( n 1) n X i =1 n X j =1 E [ X i X j ] + 1 n ( n 1) n X j =1 E [ X 2 j ] + 2 X j<k E [ X j X k ] =...
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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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