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Unformatted text preview: Confidence intervals for Means Central Limit Theorem (CLT) X 1 , X 2 , , X n : observations of a random sam ple from a distribution with mean and vari ance 2 . Then Y n = n i =1 X i n n = n ( X n ) D N (0 , 1) proof: Use MGF. (p247) Another way you may have seen the CLT: X n is approximately N ( , 2 /n ) n i =1 X i is approximately N ( n, n 2 ) 1 Example : X 1 , , X n is a random sample from Bernoulli( ). Then, Y n = n i =1 X i n q n (1 ) D N (0 , 1) Example : X Exponential( ), Y n = n i =1 X i n n . Then, Y n converges to N (0 , 1) in distribution. 2 Convergence in Distn X n : rv w/ cdf F n ( x ), n = 1 , 2 , 3 , X 1 , X 2 , converges in distribution to a rv w/ cdf F ( x ) if lim n F n ( x ) = F ( x ) for every point x at which F ( x ) is continuous and F is a cdf. Limiting MGF X n has cdf F n ( x ) and mgf M ( t ; n ) that exists for h < t < h for all n . If there exists a cdf F ( x ) with mgf M ( t ), defined for  t  h 1 < h , and lim n M ( t ; n ) = M ( t ) , then X n has a limiting distribution with cdf F ( x ). 3 Theorem U n F n ( u ) n F ( u ), V n p 1. Then W n = U n /V n has the same limiting distribution as U n ....
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 Fall '07
 Song
 Central Limit Theorem, Variance

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