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Unformatted text preview: EECS 501 CENTRAL LIMIT THEOREM Fall 2001 DEF: rvs { x 1 , x 2 . . . } are iidrv with means and variances 2 if: 1. The x i are independent : f x 1 ,x 2 ... ( X 1 , X 2 . . . ) = f x 1 ( X 1 ) f x 2 ( X 2 ) . . . 2. x i are identically distributed : f x i ( X ) = f x ( X ) , E [ x i ] = , 2 x i = 2 . THM: y n = n i =1 x i =sum of iidrvs x i with finite means and variance 2 . Then: E [ y n ] = n ; 2 y n = n 2 ; y n = y n n n = 1 n n i =1 x i = 1 n n i =1 x i . Mean: Let m n = y n n = 1 n n i =1 x i = mean . Then E [ m n ] = and 2 m n = 2 n . Proof: All of these follow immediately from the basic properties of variance. Note: Variance of (sample) mean gets smaller with n ! Regression to mean. While variance of y n grows as n , variance of y n n grows as 1 n 0. Note: Does not mean that m n remembers to correct deviations from ! DEF: The characteristic function x ( ) of rv x is x ( ) = E [ e jx ] = R  f x ( X ) e jX dX = F{ f x ( X ) } (note sign)....
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This note was uploaded on 07/22/2011 for the course EECS 370 taught by Professor Lehman during the Spring '10 term at University of Florida.
 Spring '10
 Lehman

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