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 ), , ,. .., = 12 px 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 xt ax t i i N i () = = 1 where a are arbitrary constants. The mean value i µ x and variance σ x 2 become [] µµ σµ µ xi i i N i i N ii i N i xx i i x i N i i N i Ext E aEx t a Ex t E axt a == σ =− = = = ∑∑ ∑ ∑∑ [ ( )] ( ) [ ( )] 11 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 pxd x x i j j j i j i j i i j i i j jiijj ij ij i i i i i j j j j j , ( ) ( ) [ ] ( ) −− = − = −∞ j −∞ −∞ −∞ −∞ −∞ 0 Central Limit Theorem : Under fairly common conditions, the sum random variable
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This note was uploaded on 11/21/2009 for the course ME . taught by Professor . during the Spring '09 term at Korea University.

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