51405_[Introduction to Probability and Statistics].page22-1

51405_[Introduction to Probability and Statistics].page22-1...

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1 18.05 Lecture 22 April 4, 2005 Central Limit Theorem X 1 , ..., X n - independent, identically distributed (i.i.d.) x = ( X 1 + ... + X n ) n µ = E X, θ 2 = Var( X ) n ( x µ ) n N (0 , 1) −−−−↔ θ You can use the knowledge of the standard normal distribution to describe your data: θY n ( x µ ) = Y, x µ = θ n This expands the law of large numbers: It tells you exactly how much the average value and expected vales should differ. 1 1 n ( x µ ) = n ( x 1 µ + ... + x n µ ) = n ( Z 1 + ... + Z n ) θ n θ θ where: Z i = X i π µ ; E ( Z i ) = 0 , Var( Z i ) = 1 Consider the m.g.f., see that it is very similar to the standard normal distribution: E e t 1 n ( Z 1 + ... + Z n ) = E e tZ 1 / n × ... × e tZ n / n = ( E e tZ 1 / n ) n E e tZ 1 = 1
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Unformatted text preview: + t E Z 1 + 1 t 2 E Z 1 2 + 1 t 3 E Z 1 3 + ... 2 6 1 2 1 = 1 + t + t 3 E Z 1 3 + ... 2 6 t 2 E e ( t/ n ) Z 1 = 1 + t 2 + t 3 E Z 1 3 + ... 1 + 2 n 6 n 3 / 2 2 n Therefore: ) n 2 n ( E e tZ 1 / n ) n (1 + t 2 t 2 (1 + ) n n e t / 2- m.g.f. of standard normal distribution! 2 2 n Gamma Distribution: Gamma function; for > , > 66...
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