Lect8_2700_s09 - Poisson approximation . . . Continuous RVs...

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Unformatted text preview: Poisson approximation . . . Continuous RVs Percentiles Expectation & Variance Title Page JJ II J I Page 1 of 26 Go Back Full Screen Close Quit ENGRD 2700 Basic Engineering Probability and Statistics Lecture 8: Discrete and Continuous Random Variables David S. Matteson School of Operations Research and Information Engineering Rhodes Hall, Cornell University Ithaca NY 14853 USA dm484@cornell.edu February 11, 2009 Poisson approximation . . . Continuous RVs Percentiles Expectation & Variance Title Page JJ II J I Page 2 of 26 Go Back Full Screen Close Quit 1. Poisson approximation to Binomial (cont). For n large ( n thousands), it can be difficult to compute binomial probabilities so consider the Poisson approximation: The approximation: In the binomial, suppose the success probability p and the number of trials n are such that n is large, (we learned last time that this means n 400, since otherwise minitab will give the exact answers with no trouble) p is small (say p . 01) but np is moderate (say ( np 20) . Then b ( k ; n,p ) p ( k ; = np ) . Poisson approximation . . . Continuous RVs Percentiles Expectation & Variance Title Page JJ II J I Page 3 of 26 Go Back Full Screen Close Quit Exact statement is in the form of a limit theorem: Consider a sequence of binomial models indexed by n . The n th model gives the distribution of the number of successes in n trials with success probability p ( n ) . Assume lim n p ( n ) = 0 . lim n np ( n ) = > . Then b ( k ; n,p ( n )) p ( k ; ) . Example: See last lecture. Poisson approximation . . . Continuous RVs Percentiles Expectation & Variance Title Page JJ II J I Page 4 of 26 Go Back Full Screen Close Quit Why the approximation workstheory Remember p /n : b ( k ; n,p ) = n k p k (1- p ) n- k = n ( n- 1) ... ( n- k + 1) k ! n k 1- n n- k = k k ! h n ( n- 1) ... ( n- k + 1) n n n i 1- n n 1- n- k k k ! e- . Poisson approximation . . . Continuous RVs Percentiles Expectation & Variance Title Page JJ II J I Page 5 of 26 Go Back Full Screen Close Quit Continuation of the list of mass functions to memorize. 5. Hypergeometric (from Lecture 4). 1.0.1. Hypergeometric probabilities Suppose a population of M chips labelled 1 , 2 ,...,M R , M R +1 ,...,M consists of M R red chips and M B = M- M R blue chips. Poisson approximation . . . Continuous RVs Percentiles Expectation & Variance Title Page JJ II J I Page 6 of 26 Go Back Full Screen Close Quit In a random unordered sample size n , what is the probability there are exactly k R red and k B blue chips drawn, where k R + k B = n M. Let S = all subsets of size n from population size M....
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This note was uploaded on 03/05/2009 for the course ENGRD 2700 taught by Professor Staff during the Spring '05 term at Cornell University (Engineering School).

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Lect8_2700_s09 - Poisson approximation . . . Continuous RVs...

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