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 ± 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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Suppose that we have a lake containing N N is unknown. We capture and mark m = n = 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 m ± n / 35 = 142 . If
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slides12 - Lecture Stat 302 Introduction to Probability -...

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