lec.05 - ENGG2430C Lecture 5 Discrete Random Variable...

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ENGG2430C Lecture 5: Discrete Random Variable, Expectation, and Variance Minghua Chen ([email protected]) Information Engineering The Chinese University of Hong Kong Reading: Ch. 2.1-2.4 of the textbook M. Chen ENGG2430C lecture 5 1 / 21

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Review: Total probability law (D. P. Bertsekas & J. N. Tsitsiklis, Introduction to Probability, Athena Scientific Publishers, 2002) Observing that B = B Ω = [ A 1 B ] [ A 2 B ] [ A 3 B ] and A i B ( i = 1 , 2 , 3 ) are disjoint, we get P ( B ) = P ( A 1 B )+ P ( A 2 B )+ P ( A 3 B ) = P ( A 1 ) P ( B | A 1 )+ P ( A 2 ) P ( B | A 2 )+ P ( A 3 ) P ( B | A 3 ) M. Chen ENGG2430C lecture 5 2 / 21
Review: Bayes’ law Law for combining evidences. Infer from observations. (D. P. Bertsekas & J. N. Tsitsiklis, Introduction to Probability, Athena Scientific Publishers, 2002) “Prior” probability: P ( A i ) , i = 1 , 2 , 3 For each i , we know P ( B | A i ) Compute P ( A 1 | B ) P ( A 1 | B ) = P ( A 1 B ) P ( B ) = P ( B | A 1 ) P ( A 1 ) P ( B ) (Apply total probability theorem on P ( B ) ) = P ( B | A 1 ) P ( A 1 ) j P ( A j ) P ( B | A j ) M. Chen ENGG2430C lecture 5 3 / 21

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Reviews: Counting and Bernoulli trials Permutation: order of selection matters I Choose k distinct objects out of n items, and form a sequence I n ( n - 1 ) ··· ( n - k + 1 ) = ( n k ) k ! Combination: order of selection does not matter I Choose k distinct objects out of n items, and form a group I ( n k ) Bernoulli trials: a sequence of independent and identical two-outcome stages I Toss a fair coin n times P ( we get k -heads ) = k -head sequences P ( sequence ) = (# of k -head sequences) · p k ( 1 - p ) n - k = n k p k ( 1 - p ) n - k M. Chen ENGG2430C lecture 5 4 / 21
Random Variable: Motivation Many experiments have the same probability model Sometime we are interested in functions of the outcome M. Chen ENGG2430C lecture 5 5 / 21

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Random Variable: Motivation M. Chen ENGG2430C lecture 5 6 / 21
Random Variable: Motivation M. Chen ENGG2430C lecture 5 7 / 21

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