slides12 - Lecture Stat 302 Introduction to Probability...

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Lecture Stat 302 Introduction to Probability - Slides 12 AD March 2010 AD () March 2010 1 / 32
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Hypergeometric Random Variable Consider a barrel or urn containing N balls of which m are white and N ° m are black. We take a simple random sample (i.e. without replacement) of size n and measure X , the number of white balls in the sample. The Hypergeometric distribution is the distribution of X under this sampling scheme and P ( X = i ) = ° m i ± ° N ° m n ° i ± ° N n ± AD () March 2010 2 / 32
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Example: Survey sampling Suppose that as part of a survey, 7 houses are sampled at random from a street of 40 houses in which 5 contain families whose family income puts them below the poverty line. What is the probability that: (a) None of the 5 families are sampled? (b) 4 of them are sampled? (c) No more than 2 are sampled? (d) At least 3 are sampled? Let X the number of families sampled which are below the poverty line. It follows an hypergeometric distribution with N = 40 , m = 5 and n = 7. So (a) P ( X = 0 ) (b) P ( X = 4 ) (c) P ( X ± 2 ) and (d) P ( X ² 3 ) AD () March 2010 3 / 32
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Mean and Variance of the Hypergeometric Distribution Let us introduce p = m / N then E ( X ) = np , Var ( X ) = np ( 1 ° p ) ° 1 ° n ° 1 N ° 1 ± . Suppose that m is very large compared to n , it seems reasonable that sampling without replacement is not too much di/erent than sampling with replacement. It can indeed be shown that the hypergeometric distribution can be well approximated by the binomial of parameters p = m N and N . AD () March 2010 4 / 32
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Example: Capture-Recapture Experiments We are interested in estimating the population N of animals inhabiting a certain region. To achieve this, capture-recapture studies proceed as follows. First, you capture m individuals, mark them and release them in the nature. A few days later, you capture say n animals; among the n animals some of them are marked and some are not. Let X be the number of animals which have been recaptured, then X follows an hypergeometric distribution of parameters N , m and n . Assume you have recaptured X = x animals, then you can estimate N by maximizing with respect to N the probability P ( X = x ) = ° m x ± ° N ° m n ° x ± / ° N n ± . This known as the Maximum Likelihood (ML) estimate of N . One can show that the ML estimate is the largest integer value not exceeding mn / x ; i.e. b N = b mn / x c . AD () March 2010 5 / 32
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Example: Counting Fishes Suppose that we have a lake containing N °shes where N is unknown. We capture and mark m = 100 °shes. A few days later, we capture n = 50 °shes, X of them are marked. What is the ML estimate of N if X = 35 and X = 5? If X = 35 then the ML estimate of N is b N ³ m ´ n / 35 = 142 . If X = 5 then the ML estimate of N is b N ³ m ´ n / 5 = 1000 . AD () March 2010 6 / 32
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Example: Estimating the size of an hidden population Capture-recapture ideas have been used to provide an estimate of the size of a population that cannot be directly counted. It is particularly useful for estimating the size of hidden populations for example it could be used to estimate criminal populations, victims of domestic violence or people with undiagnosed diseases.
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