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# 09_02_09_0 - 0 as n → ∞ and if ε> 4 P | X n – 3 |...

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STAT 409 Examples for 09/02/2009 Fall 2009 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 .

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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
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Unformatted text preview: 0 as n → ∞ , and if ε > 4, P ( | X n – 3 | ≥ ε ) = 0. 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 , U if 1 n n n n X = ∈ ∈ ∈ 1 , 3 2 U if 3 3 2 , 3 1 U if 2 3 1 , U if 1 Then P ( | X n – X | ≥ ε ) = n 2 , 0 < ε < 1, P ( | X n – X | ≥ ε ) = 0, ε ≥ 1. Therefore, X X P n → ....
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09_02_09_0 - 0 as n → ∞ and if ε> 4 P | X n – 3 |...

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