Lect8_2700_s09 - ENGRD 2700 Basic Engineering Probability...

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Poisson approximation . . . Continuous RV’s 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 [email protected] February 11, 2009
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Poisson approximation . . . Continuous RV’s 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 0 . 01) but np is moderate (say ( np 20) . Then b ( k ; n, p ) p ( k ; λ = np ) .
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Poisson approximation . . . Continuous RV’s 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 ) = λ > 0 . Then b ( k ; n, p ( n )) p ( k ; λ ) . Example: See last lecture.
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Poisson approximation . . . Continuous RV’s Percentiles Expectation & Variance Title Page JJ II J I Page 4 of 26 Go Back Full Screen Close Quit Why the approximation works–theory Remember p λ/n : b ( k ; n, p ) = n k p k (1 - p ) n - k = n ( n - 1) . . . ( n - k + 1) k ! λ n k 1 - λ n n - k = λ k k ! h n ( n - 1) . . . ( n - k + 1) n · n · · · · · n i 1 - λ n n 1 - λ n - k λ k k ! e - λ .
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Poisson approximation . . . Continuous RV’s 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.
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Poisson approximation . . . Continuous RV’s 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. Number of elements in S is M n . Let X represent the random variable giving X = number of red chips in the sample. What is P [ X = k R ]?
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Poisson approximation . . . Continuous RV’s Percentiles Expectation & Variance Title Page JJ II J I Page 7 of 26 Go Back Full Screen Close Quit Task: Construct a subset of size n containing exactly k R red chips and k B blue chips.
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  • Spring '05
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