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post4 - Confidence intervals for Means Central Limit...

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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 μ and vari- ance σ 2 . Then Y n = n i =1 X i - = n ( ¯ X n - μ ) σ D -→ N (0 , 1) proof: Use MGF. (p247) Another way you may have seen the CLT: ¯ X n is approximately N ( μ, σ 2 /n ) n i =1 X i is approximately N ( nμ, nσ 2 ) 1 Example : X 1 , · · · , X n is a random sample from Bernoulli( θ ). Then, Y n = n i =1 X i - q (1 - θ ) D -→ N (0 , 1) Example : X Exponential( θ ), Y n = n i =1 X i - . 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 . Example (p244) X n b ( n, p ), μ = np constant for all n M ( t ; n ) = ((1 - p ) + pe t ) n = (1 + μ ( e t - 1) n ) n -→ n →∞ exp( μ ( e t - 1)) X n D -→ Poisson( μ ) 4

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CI for Means X 1 , · · · , X n : iid N (
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