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Poisson Distribution The Poisson Distribution is a discrete distribution which takes on the values X = 0, 1, 2, 3, ... . It is often used as a model for the number of events (such as the number of telephone calls at a business or the number of accidents at an intersection) in a specific time period. It is also useful in ecological studies, e.g., to model the number of prairie dogs found in a square mile of prairie. Read Page 236 & 237 for examples Discrete Variable Distributions BINOMIAL DISTRIBUTION: Used to calculate the probability of a certain number of successes in a given number of trials POISSON DISTRIBUTION: Used to calculate probability of a number of successes that take place in a certain interval of time or space Poisson Rule for Probability e P( X r ) r! HUH ! Where did that come from? What is Lambda? Why use e? r The Quest for the Meaning of Lambda! This requires a bit of lateral thinking so hang on tight! Question What is the only function , f(x) that has 1. f(x) = f’(x) 2. f(0) = 1 ANSWER : f(x) = ex Okay try and find f’(x) of this function 2 3 n x x x x 1 .......... ...... 1! 2! 3! n! What about f(0)? It has the same properties as f(x) = e^x and if they were both graphed you can confirm that are in fact equivalent in every way GREAT BUT HOW DOES THIS HELP TO EXPLAIN WHAT LAMBDA IS ? BE PATIENT – WE’LL GET THERE WE KNOW 2 3 n x x x x e 1 .......... ...... 1! 2! 3! n! x AND e P( X r ) r! AND r P( X r ) 1 r 0 P( X r ) r 0 e 0 0! e 1 1! e 2 2! .......... 1 2 1 .......... e 1! 2! e e 1 AS IT SHOULD! So what about Lambda? Okay I’m coming to that – I said there would be some lateral thinking involved A Poisson distribution is a discrete distribution so the expected value rule still holds. X Po ( ) E ( X ) r.P( X r ) r 0 e 0 e 1 e 2 e 3 E ( X ) 0. 1. 2. 3. ... 0! 1! 2! 3! Keep going. What do you find? E(X) = LAMBDA! While we’re at it lets check variance as well Var ( X ) r 2 .P( X r ) ( E ( X )) 2 r 0 What do you find this time? Variance = Expected Value (Mean) = Lamdba! So if you are doing a problem which fits a Poisson Distribution the problem would define lambda and it could be given as the mean or as the variance (or even standard deviation). Example Top Car Rentals rents to tourists. They have 4 cars which are hired out on a daily basis. The number of requests each day occurs randomly with a mean of 3. Determine the probability that (a)None of the cars are rented (b)At least 3 of the cars are rented (c)Some requests are refused NOTE: poissonpdf and cdf are in your TI83’s DISTR Menu Yipee! ANS:a) 0.0498, b) 0.577,c) 0.185 So where did the rule come from? It came from Statisticians attempts to fit a curve to the graphed distributions shown on page 242 Look here for further evidence http://www.math.csusb.edu/faculty/stanton/m 262/poisson_distribution/Poisson_old.html