5102-Solution-Set-02

5102-Solution-Set-02 - SOLUTIONS TO PROBLEM SET 2 STAT...

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Unformatted text preview: SOLUTIONS TO PROBLEM SET 2 STAT 5102-004 (1) Suppose we are interested in the proportion θ of defective items in a large population. Consider taking a sample from that population. If the population is large compared to any sample we might see, we can safely think of the observations as independent Bernoulli trials with common probability θ that a defective item is drawn. Problem 6.2.6 describes sampling a fixed number n of items from the population and counting the number of defectives. This sampling is described by a binomial distribution. The distribution of the number of defective items Y has density f ( y | θ ) = n y θ y (1- θ ) n- y θ ∈ (0 , 1) n ∈ N y ∈ { ,...,n } . Now, assume that we have a uniform prior for the proportion of defectives. The posterior density is proportional to the likelihood times the prior density. In this case, lik( θ | y ) ∝ θ y (1- θ ) n- y and π ( θ ) ∝ 1, so that the posterior density is proportional to θ ( y +1)- 1 (1- θ ) ( n- y +1)- 1 θ ∈ (0 , 1) . We recognize this as the kernel of a beta density with α = y + 1 and β = n- y + 1. Since y = 3 of the n = 8 observations are defectives, the posterior for θ is Beta(4 , 6). Problem 6.2.9 describes sampling from the population until a fixed number r of defective items are observed. This sampling is described by a negative binomial distribution. The distribution of the number of non-defective items X has density f ( x | θ ) = x + r- 1 x θ r (1- θ ) x θ ∈ (0...
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This note was uploaded on 02/24/2011 for the course STAT 5102 taught by Professor Staff during the Spring '03 term at Minnesota.

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5102-Solution-Set-02 - SOLUTIONS TO PROBLEM SET 2 STAT...

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