There are two complementary approaches for dealing

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Unformatted text preview: 2 3 + X2 2 = 4 9 + E [X 2 ] 4 = 35 72 . MSE for L∗ (X ) = Thus, 1 35 1 − =. 2 72 72 1 Note that (MSE using g ∗ (X ) = 96 ) ≤ (MSE using L∗ = are ordered in accordance with (4.36). 4.11 1 72 ) ≤ (Var(Y ) = 1 18 ), so the three MSEs Joint Gaussian distribution Recall that Gaussian distributions often arise in practice; this fact is explained by the CLT. The CLT can be extended to two, or even more, correlated random variables. For example, suppose (U1 , V1 ), (U2 , V2 ), · · · are independent, identically distributed pairs of random variables. For example, Ui might be the height, and Vi the weight, of the ith student to enroll at a university. Suppose ···+ ···+ for convenience that they have mean zero. Then as n → ∞, the pair U1 +√n Un , V1 +√n Vn has a limiting bivariate distribution, where “bivariate” means the limit distribution is a joint distribution of two random variables. Suppose X and Y have such a limit distribution. Then X and Y must each be Gaussian random var...
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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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