We will focus on the later one in this course jimin

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We will focus on the later one in this course. Jimin Ding, Math WUSTL Math 494 Spring 2018 4 / 44
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Example 1: Quality Control Consider a population of N elements, for instance, a shipment of manufactured items. An unknown number of these elements are defective. It will be expensive to exam all items for large N (or impossible if the inspection is destructive). To learn θ , one may randomly draw a sample of n without replacement and inspect. (Assume all items have the same probability to be defective.) I Population: I Sample (data): I Probability model: Jimin Ding, Math WUSTL Math 494 Spring 2018 5 / 44
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Example 1: Quality Control Consider a population of N elements, for instance, a shipment of manufactured items. An unknown number of these elements are defective. It will be expensive to exam all items for large N (or impossible if the inspection is destructive). To learn θ , one may randomly draw a sample of n without replacement and inspect. (Assume all items have the same probability to be defective.) I Population: N items, defective items I Sample (data): I Probability model: Jimin Ding, Math WUSTL Math 494 Spring 2018 5 / 44
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Example 1: Quality Control Consider a population of N elements, for instance, a shipment of manufactured items. An unknown number of these elements are defective. It will be expensive to exam all items for large N (or impossible if the inspection is destructive). To learn θ , one may randomly draw a sample of n without replacement and inspect. (Assume all items have the same probability to be defective.) I Population: N items, defective items I Sample (data): n sampled items, X defective items in sample I Probability model: Jimin Ding, Math WUSTL Math 494 Spring 2018 5 / 44
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Example 1: Quality Control Consider a population of N elements, for instance, a shipment of manufactured items. An unknown number of these elements are defective. It will be expensive to exam all items for large N (or impossible if the inspection is destructive). To learn θ , one may randomly draw a sample of n without replacement and inspect. (Assume all items have the same probability to be defective.) I Population: N items, defective items I Sample (data): n sampled items, X defective items in sample I Probability model: Pr ( X = k ) = ( k )( N - n - k ) ( N n ) , k = 0 , 1 , · · · , min( Nθ, n ) . This is the hypergeometric distribution, H ( Nθ, N, n ) , which is a family of distributions indexed by θ . Here θ is the unknown parameter that we want to estimate. Jimin Ding, Math WUSTL Math 494 Spring 2018 5 / 44
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Example 2: Measurement Problem An experimenter makes n independent determinations of a physical constant μ . I Population: I Sample (data): I Probability model: Jimin Ding, Math WUSTL Math 494 Spring 2018 6 / 44
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Example 2: Measurement Problem An experimenter makes n independent determinations of a physical constant μ .
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