# 03_26 - STAT 410 Examples for 03/26/2008 Spring 2008 Let...

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Examples for 03/26/2008 Spring 2008 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 5 : Let X n have p.d.f. f n ( x ) = n n x 2 1 1 1 + + , for 0 < x < 1, zero elsewhere. Then X X D n , where X has a Uniform distribution over ( 0, 1 ). Example 6 : Let X have a Uniform distribution over ( 0, 1 ). Let P ( X n = n i ) = n 1 , for i = 1, 2, … , n . Note that X is continuous, while X n ’s are discrete. For 0 < x < 1, F X ( x ) = x , F X n ( x ) = n x n , where x ± = the greatest integer less than or equal to x . Therefore,

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## This note was uploaded on 02/11/2011 for the course STAT 410 taught by Professor Monrad during the Spring '08 term at University of Illinois, Urbana Champaign.

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03_26 - STAT 410 Examples for 03/26/2008 Spring 2008 Let...

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