MIT14_30s09_lec15

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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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14.30 Introduction to Statistical Methods in Economics Lecture Notes 15 Konrad Menzel April 7, 2009 1 Special Distributions (continued) 1.1 The Poisson Distribution Often, we may be interested in how often a certain event occurs in a given interval. Example 1 In airline safety, we may want to have some notion of how ”safe” an airplane model is. The following data are from the website www.airsafe.com , and give the total number of ﬂights and number of fatal events involving aircraft of a given type up to December 2006. Model Flights Events Airbus A300 10.35M 9 Airbus A310 3.94M 6 Airbus A320/319/321 30.08M 7 Airbus A340 1.49M 0 Boeing 727 76.40M 48 Boeing 737 127.35M 64 Boeing 747 17.39M 28 Boeing 757 16.67M 7 Boeing 767 13.33M 6 Boeing 777 2.0M 0 Boeing DC9 61.69M 43 Boeing DC10 8.75M 15 Boeing MD11 1.69M 3 Boeing MD80/MD90 37.27M 14 Concorde 0.09M 1 We can see immediately that some types of aircraft have had fewer accidents than others simply because they haven’t been in service for long or were only produced in small numbers. In order to be able to make a more meaningful comparison, we need a better way of describing the distribution of the number of fatal events. Random variables of this type are commonly referred to as count data , and an often used distribution to describe them is the Poisson distribution 1
Definition 1 X is said to follow Poisson distribution with arrival rate λ , X P ( λ ) , if it has the p.d.f. λ f X ( x ) = λ x x e ! if x ∈ { 0 , 1 , 2 , . . . } 0 otherwise Note that in particular, X is discrete. Property 1 For a Poisson random variable X , E [ X ] = λ Var( X ) = λ In order to see why the Poisson distribution is a plausible candidate for the distribution of a count variable, let’s do the following thought experiment: Suppose the probability that the event happens in a time window of length n 1 is p n = n λ . we also assume that the event happening at any given instant is independent across time. We then let the partition of subintervals grow finer by letting n go to infinity. If the probability of two occurrences in the same shrinking subinterval goes to zero, and we then count the number of subintervals in which the event occurs at least once we get the total number of occurrences. Notice that this will be a binomial random variable with parameters p = n λ and n . Proposition 1 For the binomial random variable X n B n, n λ , as n → ∞ , the p.d.f. converges to n λ x λ n x e λ λ x lim f X n ( x ) = lim 1 = n n x n n x ! Proof: We can take the limit of the product as the product of limits, and evaluate each term separately: By a well-known result from calculus (e.g. can do Taylor expansions on both sides), n λ lim 1 = e λ n n So we are left with the term �� � x x x n λ λ n ! λ T n = 1 = x n n x !( n x )! n λ for which we can show n ( n 1) . . . ( n k + 1) λ x λ x n n 1 n x + 1 λ x lim T n = lim =

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