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08CentralLimit

08CentralLimit - MAE 591 RANDOM DATA Central Limit Theorem...

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MAE 591 RANDOM DATA Central Limit Theorem Let x t i N i ( , be N statistically independent random variables with respective probability densities ), , ,..., = 1 2 p x i ( ) . Let µ i and σ i 2 be the mean value and variance of each random variable x t i ( ) . Consider the sum random variable x t a x t i i N i ( ) ( ) = = 1 where a are arbitrary constants. The mean value i µ x and variance σ x 2 become ( ) [ ] ( ) µ µ σ µ µ x i i i N i i N i i i N i x x i i x i N i i N i E x t E a x t a E x t a E x t E a x t a = = σ = = = = = = = = = = [ ( )] ( ) [ ( )] ( ) ( ) 1 1 2 2 1 2 2 1 2 1 Proof : ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) E a a x t x t a a x t x t p x x dx dx a a x t x t p x p x dx dx independency a a x t p x dx x t p x dx i j i i j j i j i i j j i j i i j i i j j i i j j i j i j i i i i i j j j j j ( ) ( ) ( ) ( ) , ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) = = = = −∞ j −∞ −∞ −∞ −∞ −∞
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