Example_class_4_handout2

# Example_class_4_handout2 - THE UNIVERSITY OF HONG KONG...

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THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 4 Review Abbreviated event notation Usually we abbreviate the description of the event or the set { ω : X ( ω ) x } to simply as { X x } so that we can also abbreviate the probability notation P ( { ω : X ( ω ) x } ) as P ( { X x } ) or even simpler as P ( X x ). This is primarily because the two probability measures P X and P are consistent due to the property of the random variable X . Deﬁnition of CDF F X ( x ) = P ( X x ) = P ( { ω : X ( ω ) x } ), where x ( -∞ , + ) Properties of CDF (a) 0 F X ( x ) 1 (b) If x 1 < x 2 , then F X ( x 1 ) F X ( x 2 ). (c) lim x + F X ( x ) = F X (+ ) = 1 (d) lim x →-∞ F X ( x ) = F X ( -∞ ) = 0 (e) lim x a + F X ( x ) = F X ( a + ) = F X ( a ), where a + = lim 0 0 ( a + ± ) (f) P ( a < X b ) = F X ( b ) - F X ( a ) (g) P ( X > a ) = 1 - F X ( a ) (h) P ( X < b ) = F X ( b - ) , where b - = lim 0 0 ( b - ± ) Deﬁnition of PMF p X ( x ) = P ( X = x ) = P ( { ω : X ( ω ) = x } ), where x ( -∞ , + ) 1

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Properties of PMF The state space of a discrete random variable X must be countable. Denote the state space of a discrete r.v. X by S = { x 1 ,x 2 ,x 3 ,... } . (a) 0 p X ( x k ) 1, k = 1 , 2 ,... (b) If x 6 = x k for all k = 1 , 2 ,... , then p X ( x ) = 0 (c) X k p X ( x k ) = 1 (d) F X ( x ) = x k x p X ( x k ) Remark. P ( E ) = 0 ; E = φ Remark. If X is a continuous random variable, then
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Example_class_4_handout2 - THE UNIVERSITY OF HONG KONG...

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