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Unformatted text preview: X ( x ) = & & & ± & & & ² ³ ≥ < ≤ < 1 1 1 2 1 x x x ( ) ( ) x x n n X X F F lim = ∞ → , for all x ≠ 0. F X n ( ) = 0 for all n , but F X ( ) = 2 1 . ( ) ( ) F 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 F X n ( x ) = & & & ± & & & ² ³ ≥ < ≤ < 1 1 1 x x x x n . ( ) & ± & ² ³ ≥ < ∞ = → 1 1 1 F X lim x x x n n . Therefore, X X D n → , where P ( X = 1 ) = 1. Note 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....
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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.
 Spring '08
 Monrad

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