Lecture_6-2_Aug_2010

Lecture_6-2_Aug_2010 - EE 131A Probability Professor Kung...

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UCLA EE131A (KY) 1 EE 131A Probability Professor Kung Yao Electrical Engineering Department University of California, Los Angeles M.S. On-Line Engineering Program Lecture 6-2
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UCLA EE131A (KY) 2 Cumulative Distribution Function (5) Ex. 1. Toss a coin. Sample space S = {H, T}; P(H) = 0.6 and P(T) = 0.4 . Let the rv X = -5, for the outcome of a “tail” and X = 3 for the outcome of a “head.” Then the induced sample space S X = {-5, 3} and P(X = -5) = 0.4 and P(X = 3) = 0.6 . The cdf F(x) is given by We note this cdf takes values on [0, 1]. As x - , F(x) 0. As x  , F(x) 1. F(x) is a non- decreasing function. At F(-5 - ) = 0, F(-5) = F(-5 + ) = 0.4, while at F(3 - ) = 0.4, F(3) = F(3 + ) = 1. -5 3 0 x .4 .6 F(x) 0 1 1, 3 x< , F(x) = .4, -5 x<3, 0,- <x<-5.
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UCLA EE131A (KY) 3 Cumulative Distribution Function (6) 6. P(a < X b) = F(b) – F(a) . (*) To show (*), note {X a} {a < X b} = {X b}. Since the two events on the lhs are mutually exclusive, then F(a) + P(a < X b) = F(b) . 7. P(X = b) = F(b) – F(b - ). (**) To show (**), note {X < b} {X = b} = {X b}. Since the two events on the lhs are mutually exclusive, then
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Lecture_6-2_Aug_2010 - EE 131A Probability Professor Kung...

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