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# stat400lec23 - Statistics 400 Section 6.5 and 6.6 Central...

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Statistics 400 Section 6.5 and 6.6 Central limit theorem Central Limit Theorem (CLT) If X1, X2, …, Xn are observations of a random sample of size n from a distribution with mean μ and variance σ 2 , Then we have W= n X / σ μ - = n n X i σ μ - N(0,1) as n →∞ In another word, W be can approximated by normal distribution when sample size n is sufficiently large. “sufficiently large”: n is greater than 20 Approximation of Poisson distribution If X1, X2, …, Xn are observations of a random sample of size n from the Poisson distribution with mean 1, Y= = n i i X 1 is Poisson distributed with mean n. W= n n X i σ μ - = n n Y - = n X / 1 1 - N(0,1) “sufficiently large”: n 20 and P(Y k) Φ ( n n k - + 5 . 0 ) Ping Ma Lecture 23 Fall 2005 - 1 -

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Example: Let Y have a Poison distribution with mean 20. Calculate P(16<Y 21) and approximate it using CLT Ping Ma Lecture 23 Fall 2005 - 2 -
If X1, X2, …, Xn are observations of a random sample of size n from a normal distribution N( μ , σ 2 ), then we have (a) X is N( μ , σ 2 /n) n X / σ μ

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