10_14_2 - STAT 410 Examples for 07/15/2011 (2) Summer 2011...

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Examples for 07/15/2011 (2) Summer 2011 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 ( ) ( ) 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: Let X n have p.d.f. f n ( x ) = n x n – 1 , for 0 < x < 1, zero elsewhere. Then F X n ( x ) = < < 1 1 1 0 0 0 x x x x n . ( ) < = 1 1 1 0 F X lim x x x n n . Therefore, X X D n , where P ( X = 1 ) = 1. Recall that 1 X P n , since if 0 < ε 1, P ( | X n – 1 | ε ) = ( 1 – ε ) n 0 as n , and if ε > 1, P ( | X n – 1 | ε ) = 0. Example 2: Let X n have p.d.f. f n ( x ) = n e n x , for x > 0, zero otherwise. Then
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This note was uploaded on 10/31/2011 for the course MATH 464 taught by Professor Monrad during the Fall '08 term at University of Illinois, Urbana Champaign.

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10_14_2 - STAT 410 Examples for 07/15/2011 (2) Summer 2011...

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