# chapter7 - Approximations to Probability Distributions...

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Approximations to Probability Distributions: Limit Theorems

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Sequences of Random Variables Interested in behavior of functions of random variables such as means, variances, proportions For large samples, exact distributions can be difficult/impossible to obtain Limit Theorems can be used to obtain properties of estimators as the sample sizes tend to infinity Convergence in Probability – Limit of an estimator Convergence in Distribution – Limit of a CDF Central Limit Theorem – Large Sample Distribution of the Sample Mean of a Random Sample
Convergence in Probability The sequence of random variables, X 1 ,…,X n , is said to converge in probability to the constant c , if for every ε >0, Weak Law of Large Numbers (WLLN): Let X 1 ,…,X n be iid random variables with E(X i )= μ and V(X i )= σ 2 < . Then the sample mean converges in probability to μ : 1 ) | (| lim = - ε c X P n n ( 29 ( 29 n X X X P X P n i i n n n n n = = = - = - 1 where 1 lim or 0 lim μ

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( 29 ( 29 ( 29 μ ε σ Prob 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 0 0 lim | | lim 1 | | 1 : Let 1 | | 1 ) | (| 1 1 ) | (| ) 1 ( 1 1 ) ( : Inequality s Chebyshev' 2200 = - = = = - = = = = = - - - - - + - = = = = n n X n n X X n X X n X X X X X X X X X n X n X k n k n k n k k n k k X P k k X P k k X P k k k X k P n n X V X E
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## This note was uploaded on 06/04/2011 for the course STA 4321 taught by Professor Staff during the Spring '08 term at University of Florida.

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chapter7 - Approximations to Probability Distributions...

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