07_17 - STAT 410 Examples for 07/17/2009 Summer 2009 Let...

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STAT 410 Examples for 07/17/2009 Summer 2009 Let { X n } be a sequence of random variables and let X be a random variable. Let F X n and F X be, respectively, the c.d.f.s of X n and X. Let C ( F X ) denote the set of all points where F X is continuous. We say that X n converges in distribution to X if ( 29 ( 29 x x n n X X F F lim = , for all x C ( F X ). We denote this convergence by X X D n . Example 1 : Consider { X n } with p.m.f.s P ( X n = 3 ) = 1 – n 1 , P ( X n = 7 ) = n 1 . Note that 3 X P n , since if 0 < ε < 4, P ( | X n – 3 | ε ) = n 1 0 as n , and if ε 4, P ( | X n – 3 | ε ) = 0. Consider X with p.m.f. P ( X = 3 ) = 1. F X n ( x ) = < - < 7 1 7 3 1 1 3 0 x x n x F X ( x ) = < 7 1 3 0 x x ( 29 ( 29 x x n n X X F F lim = , for all x R . X X D n . Example 2 : Let { X n }, X be i.i.d. with p.m.f. P ( X n = n 1 ) = n 1 2 1 - , P ( X n = 1 ) = n 1 2 1 + . Then F X n ( x ) = < - < 1 1 1 1 1 2 1 1 0 x x n n n x . ( 29 < < = 1 1 1 0 2 1 0 0 F X lim x x x x n n .
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Consider X with p.m.f. P ( X = 0 ) = 2 1 , P ( X = 1 ) = 2 1 . Then F X ( x ) = < < 1 1 1 0 2 1 0 0 x x x ( 29 ( 29 x x n n X X F F lim = , for all x 0. F X n ( 0 ) = 0 for all n , but F X ( 0 ) = 2 1 . ( 29 ( 29 0 F 0 F X X lim n n . Since 0 C ( F X ), X X D n . Example 3 : Suppose P ( X n = i ) = 6 3 + + n i n , for i = 1, 2, 3. Then X X D n , where P ( X = i ) = 3 1 , for i = 1, 2, 3. Example 4 : Let X n have p.d.f. f n ( x ) = n x n – 1 , for 0 < x < 1, zero elsewhere. Then
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This note was uploaded on 10/29/2009 for the course STAT 410 taught by Professor Alexeistepanov during the Summer '08 term at University of Illinois at Urbana–Champaign.

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07_17 - STAT 410 Examples for 07/17/2009 Summer 2009 Let...

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