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**Unformatted text preview: **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 . ) Ping Ma Lecture 23 Fall 2005- 1 - 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...

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