# Either the integral in 427 may not have a closed form

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Unformatted text preview: X ] = 1 n n E [Xk ] = k=1 1 n n µ = µ, k=1 so X is an unbiased estimator of µ. (b) The mean square error for estimating µ by X is given by E [(µ − X )2 ] = Var(X ) = 1 Var n2 n Xk = k=1 n 1 n2 Var(Xk ) = k=1 1 n2 n σ2 = k=1 σ2 . n (c) E σ2 = 1 n−1 because, by symmetry, E [(Xk − n E [(Xk − X )2 ] = k=1 X )2 ] E (X1 − X )2 n E [(X1 − X )2 ], n−1 = E [(X1 − X )2 ] for all k. Now, E [X1 − X ] = µ − µ = 0, so = Var(X1 − X ) = Var = (n − 1)X1 − n n−1 n n 2 + k=2 n k=2 1 n2 Xk n σ2 = (n − 1)σ 2 . n Therefore, E σ2 = n (n − 1)σ 2 = σ2, n−1 n so, σ 2 is an unbiased estimator of σ 2 . 4.9 Law of large numbers and central limit theorem The law of large numbers, in practical applications, has to do with approximating sums of random variables by a constant. The Gaussian approximation, backed by the central limit theorem, in practical applications, has to do with a more reﬁned approximation: approximating sums of random variables by a single Gaussian random variable. 4.9. LAW OF LARGE NUMBE...
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## This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Institute of Technology.

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