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sta257week4notes - Relation between Binomial and Poisson...

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week 4 1 Relation between Binomial and Poisson Distributions Binomial distribution Model for number of success in n trails where P (success in any one trail) = p . Poisson distribution is used to model rare occurrences that occur on average at rate λ per time interval. Can think of “rare” occurrence in terms of p Æ 0 and n Æ . Take these limits so that λ = np . So we have that

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week 4 2 Continuous Probability Spaces is not countable. Outcomes can be any real number or part of an interval of R , e.g. heights, weights and lifetimes. Can not assign probabilities to each outcome and add them for events. Define as an interval that is a subset of R . F – the event space elements are formed by taking a (countable) number of intersections, unions and complements of sub-intervals of . Example: = [0,1] and F = { A = [0,1/2), B = [1/2, 1], Φ , }
week 4 3 How to define P ? Idea - P should be weighted by the length of the intervals. - must have P ( ) = 1 - assign 0 probability to intervals not of interest. •F o r the real line, define P by a (cumulative) distribution function as follows: F ( x ) = P ((- , x ]). Distribution functions (cdf) are usually discussed in terms of random variables.

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week 4 4 Recalls
week 4 5 Cdf for Continuous Probability Space For continuous probability space, the probability of any unique outcome is 0. Because, P ({ ω }) = P (( ω , ω ]) = F ( ω ) - F ( ω ) = 0.

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This note was uploaded on 08/27/2008 for the course STA 257 taught by Professor Hadasmoshonov during the Summer '08 term at University of Toronto.

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sta257week4notes - Relation between Binomial and Poisson...

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