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# 11clt - EECS 501 CENTRAL LIMIT THEOREM Fall 2001 DEF rvs...

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EECS 501 CENTRAL LIMIT THEOREM Fall 2001 DEF: rvs { x 1 , x 2 . . . } are iidrv with means μ and variances σ 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 ] = μ, σ 2 x i = σ 2 . THM: y n = n i =1 x i =sum of iidrvs x i with finite means μ and variance σ 2 . Then: E [ y n ] = ; σ 2 y n = 2 ; ˜ y n = y n - = 1 n n i =1 x i - μ σ = 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 ] = μ and σ 2 m n = σ 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 μ ! DEF: The characteristic function Φ x ( ω ) of rv x is Φ x ( ω ) = E [ e jωx ] = R -∞ f x ( X ) e jωX dX = F{ f x ( X ) } (note sign).

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