10_17 - STAT 410 Example 1: Examples for 10/17/2011 Fall...

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STAT 410 Examples for 10/17/2011 Fall 2011 Example 1: Let X 1 , X 2 , … be i.i.d. with mean μ and standard deviation σ . Let n n n X ... X X 1 + + = , n = 1, 2, … . We already know that μ X P n . Then for all ε > 0, ( ) X P ε μ - n 0, ( ) X P ε μ < - n 1 as n . We wish to show that μ X D n . F μ ( x ) = < μ μ 1 0 x x Since F μ ( x ) is not continuous at μ , we need to show o if x < μ , ( ) x n n X F lim = 0, t if x > μ , ( ) x n n X F lim = 0. o If x < μ , then ε > 0 such that x μ ε . Then ( ) x n X F ( ) X P ε μ - n ( ) X P ε μ - n 0 as n . t If x > μ , then ε > 0 such that x μ + ε . Then ( ) x n X F ( ) X P ε μ + n ( ) X P ε μ < - n 1 as n . Therefore, μ X D n . Theorem 1 X X P n X X D n
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Example 2(a): Let { X n }, X be i.i.d. with p.m.f. P ( X = – 1 ) = 2 1 , P ( X = 1 ) = 2 1 . Then
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10_17 - STAT 410 Example 1: Examples for 10/17/2011 Fall...

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