CentralLimitTheorem-4

# CentralLimitTheorem-4 - Weak Law of Large Numbers Strong...

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1 Weak Law of Large Numbers Consider I.I.D. random variables X 1 , X 2 , . .. X i have distribution F with E[X i ] = m and Var(X i ) = s 2 Let For any e > 0: Proof: By Chebyshev’s inequality: 0 ) (   n X P e n n X X X E X E ... 2 1 ] [ n n n X X X X 2 ... 2 1 Var ) ( Var 0 ) ( 2 2   n n X P n i i X n X 1 1 Strong Law of Large Numbers Consider I.I.D. random variables X 1 , X 2 , . .. X i have distribution F with E[X i ] = Let Strong Law Weak Law, but not vice versa Strong Law implies that for any > 0, there are only a finite number of values of n such that condition of Weak Law: holds. n i i X n X 1 1 1 ... lim 2 1 n X X X P n n X Intuitions and Misconceptions of LLN Say we have repeated trials of an experiment Let event E = some outcome of experiment Let X i = 1 if E occurs on trial i , 0 otherwise Strong Law of Large Numbers (Strong LLN) yields: Recall first week of class: Strong LLN justifies “frequency” notion of probability Misconception arising from LLN: o Gambler’s fallacy: “I’m due for a win” o Consider being “due for a win” with repeated coin flips. .. ) ( ] [ ... 2 1 E P X E n X X X n n E n E P n ) ( lim ) ( La Loi des Grands Nombres History of the Law of Large Numbers 1713: Weak LLN described by Jacob Bernoulli 1835: Poisson calls it “La Loi des Grands Nombres”

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## This document was uploaded on 12/24/2011.

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CentralLimitTheorem-4 - Weak Law of Large Numbers Strong...

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