MIT14_30s09_lec15

MIT14_30s09_lec15 - MIT OpenCourseWare http/ocw.mit.edu...

Info icon This preview shows pages 1–4. Sign up to view the full content.

View Full Document Right Arrow Icon
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 .
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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
Image of page 2
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 =
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern