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Unformatted text preview: ≥ n b b b 2 1 Proof Let . Since n n b b ∞ → = lim { } 1 , ≥ n b n is decreasing, the sets , represent a decreasing sequence of events, with lim . Using the continuity property of probability, we have ( ] n i b E , ∞ − = ( ] b E i i , 1 ∞ − = ∞ = 1 ≥ i E n n = ∞ → I ( ) ( ) ( ) ( ) ( ) ( ) ( n n n n n n i i n n b F b X P E P E P b X P b F b F ∞ → ∞ → ∞ → ∞ = ∞ → = ≤ = = = ≤ = = lim lim lim lim 1 I ) } . Note that if { is increasing, then 1 , ≥ n b n ( ] n i b E , ∞ − = , would represent an increasing sequence of events, and . Therefore 1 ≥ i ( b , ∞ ) E i i 1 − = = ∞ = U E n n lim ∞ → ( ) ( ) ( ) ( ) ( b F b X P E P E P b F i i n n n n ≠ < = = = ∞ = ∞ → ∞ → U 1 lim lim ) ....
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This note was uploaded on 05/04/2011 for the course STAT 1301 taught by Professor Smslee during the Spring '08 term at HKU.
 Spring '08
 SMSLee
 Probability

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