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Unformatted text preview: X ] = 1
n n E [Xk ] =
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
n2 n Xk = k=1 n 1
n2 Var(Xk ) =
n2 n σ2 =
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 2 +
σ2 = (n − 1)σ 2
E σ2 = n (n − 1)σ 2
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.
- Spring '08
- The Land