Week_10_Slides - Week 10 Central Limit Theorem Point...

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Week 10 Central Limit Theorem Point Estimation
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Lindeberg-Levy Central Limit Theorem (CLT) Suppose that 1 2 , ,... X X are iid rv’s with finite mean , finite variance 2 and a mgf. Let 1 1 n n i i X X n and / n n X U n . Then N(0,1) d n U   .
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Steps to Prove the CLT
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The Uniqueness Theorem Suppose that 1 2 , , ,... X X X are rv’s with cdf’s 1 2 ( ), ( ), ( ),. .. F x F x F x , respectively, an d mgf’s 1 2 ( ), ( ), ( ),. .. m t m t m t , respectively, such that ( ) ( ) n m t m t as n   t    . Then ( ) ( ) n F x F x x    .
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Variance Estimated CLT N(0,1) / d n n n X W S n   as n  
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Suppose that 1 2 , ,... X X are iid rv’s with finite mean , finite variance 2 and a mgf. Let 1 1 n n i i X X n (sample mean) and 2 2 1 1 ( ) 1 n n i i S X X n (sample variance). n  
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Week_10_Slides - Week 10 Central Limit Theorem Point...

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