10_12 - STAT 410 Examples for 10/12/2011 Fall 2011 Def Let...

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STAT 410 Examples for 10/12/2011 Fall 2011 Def Let U 1 , U 2 , … be an infinite sequence of random variables, and let W be another random variable. Then the sequence { U n } converges in probability to W, if for all ε > 0, ( ) 0 W U P lim ε = - n n , and write W U P n . Example 1: Let X n have p.d.f. f n ( x ) = n x n – 1 , for 0 < x < 1, zero otherwise. Then 1 X P n , since if 0 < ε 1, P ( | X n – 1 | ε ) = 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 0 X P n , since if ε > 0, P ( | X n – 0 | ε ) = P ( X n ε ) = e n ε 0 as n . Example 3: Let X n have p.m.f. P ( X n = 3 ) = 1 – n 1 , P ( X n = 7 ) = n 1 . Then 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.
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Example 4: Suppose U ~ Uniform ( 0, 1 ). Let X n = - - + + 1 , 1 3 2 U if 3 1 3 2 , 1 3 1 U if 2 1 3 1 , 0 U if 1 n n n n X = 1 , 3 2 U if
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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_12 - STAT 410 Examples for 10/12/2011 Fall 2011 Def Let...

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