Example class 3 handout

Example class 3 handout - THE UNIVERSITY OF HONG KONG...

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Unformatted text preview: THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1301 PROBABILITY AND STATISTICS I EXAMPLE CLASS 3 Review Probability (a) Law of Total Probability (i) If 0 < P ( B ) < 1, then P ( A ) = P ( A | B ) P ( B ) + P ( A | B c ) P ( B c ) for any A. (ii) If B 1 ,B 2 ,...,B k are mutually exclusive and exhaustive events (i.e. a partition of the sample space), then for any event A, P ( A ) = k ∑ j =1 P ( A | B j ) P ( B j ) where k can also be ∞ . (b) Bayes’ Theorem (Bayes’ rule, Bayes’ law) For any two event A and B with P ( A ) > 0 and P ( B ) > 0, P ( B | A ) = P ( A | B ) P ( B ) P ( A ) (c) Bayes’ Theorem If B 1 ,B 2 ,...,B k are mutually exclusive and exhaustive events (i.e. a partition of the sample space), and A is any event with P ( A ) > , then for any B j , P ( B j | A ) = P ( A | B j ) P ( B j ) P ( A ) = P ( B j ) P ( A | B j ) ∑ k i =1 P ( B i ) P ( A | B i ) where k can also be ∞ . Random variables (a) Basics- Random variable is a measurable function between a Sample Space (Domain) and State Space (Range).- A function (with some requirement) X : Ω ∋ ω 7→ X ( ω ) ∈ X (Ω) defined on the sample space Ω = { ω } is called a random variable. 1 (b) Distribution- Law of a Random Variable’s Dance There is a law governing how any random variable to be observed in its state space. The law is a probabilistic one, called the probability distribution of a random variable. There are two qualifications for any real-valued function...
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This note was uploaded on 01/16/2012 for the course STAT 1301 taught by Professor Smslee during the Fall '08 term at HKU.

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Example class 3 handout - THE UNIVERSITY OF HONG KONG...

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