elemprob-fall2010-page44

Elemprob-fall2010-pa - Theorem 17.4 Suppose the Xi are i.i.d Suppose E Xi2 < Let = E Xi and 2 = Var Xi Then P a Sn n b P(a Z b n for every a and b

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Theorem 17.4 Suppose the X i are i.i.d. Suppose E X 2 i < . Let μ = E X i and σ 2 = Var X i . Then P ± a S n - σ n b ² P ( a Z b ) for every a and b , where Z is a N (0 , 1) . The ratio on the left is ( S n - E S n ) / Var S n . We do not claim that this ratio converges for any ω (in fact, it doesn’t), but that the probabilities converge. An example. If the X i are i.i.d. Bernoulli random variables, so that S n is a binomial, this is just the normal approximation to the binomial. An example. Suppose we roll a die 3600 times. Let X i be the number showing on the i th roll. We know S n /n will be close to 3 . 5. What’s the probability it differs from 3 . 5 by more than 0 . 05? Answer. We want P ±³
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This note was uploaded on 12/29/2011 for the course MATH 316 taught by Professor Ansan during the Spring '10 term at SUNY Stony Brook.

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