2.2.1 Continuity of Probability and CDF

2.2.1 Continuity of Probability and CDF - ≥ n b b b 2 1...

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Continuity of Probability Measure For an increasing sequence of event , define . L L n E E E 2 1 U = = 1 lim i i n n E E For a decreasing sequence of event L L n E E E 2 1 , define . I = = 1 lim i i n n E E If is an increasing (or decreasing) sequence of events, then { 1 , n E n } ( ) ( ) n n n n E P E P = lim lim . Proof Suppose { is increasing, then we can let } } 1 , n E n 1 1 E F = , , , , c E E F 1 2 2 = c E E F 2 3 3 = c n n n E E F 1 = so that { are mutually exclusive and , . 1 , n F n n n i i E F = = U 1 U U = = = 1 1 i i i i E F According to the definition of the limit of n E ( ) ( ) U = = 1 lim i i n n E P E P ( ) U = = 1 i i F P ( 3 rd axiom) () = = 1 i i F P () = = n i i n F P 1 lim ( ) U n i i n F P 1 lim = = (3 rd axiom) () n n E P = lim The proof for decreasing sequence of events is left as exercise.
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Continuity of Cumulative Distribution Function For any random variable X , the cumulative distribution function defined by () ( ) x X P x F = is right continuous everywhere, i.e. ( ) ( ) n n n n b F b F = lim lim for any decreasing sequence of numbers . L L
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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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2.2.1 Continuity of Probability and CDF - ≥ n b b b 2 1...

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