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# Week03Notes - Stat 311 Introduction to Mathematical...

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Stat 311 Introduction to Mathematical Statistics Zhengjun Zhang Department of Statistics, University of Wisconsin, Madison, WI 53706, USA September 15-17, 2009 Mutually exclusive sets: No common element between any pair of sets. Exhaustive sets: The union of all sets is the sample space Ω. A and ˜ A are mutually exclusive and exhaustive. Long run relative frequency Sometimes there is no reason to believe outcomes are equally likely. Process: repeat an experiment over and over and count how many times an event A occurs. Assume we run N trials, then P ( A ) # of times A occurs N Benford’s law The first digits of numbers in legitimate records often follow a distribution First digit 1 2 3 4 5 6 7 8 9 Proportion .301 .176 .125 .097 .079 .076 .058 .051 .046 Example Want to find the probability of new car defects Sample a group of 237,412 new car owners Find 2506 complains about their vehicles A = { a car has defects } P ( A ) # complains # surveys = 2506 237412 = 0 . 0106 non equally likely, non long run relative frequency 0

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Need prior information Ex: Two friends are both in love with a woman, but only one (maybe none) can marry the woman. Bayesian procedure? Odds Determine your own personal probability using your prior knowledge. The odds in favor of an event or a proposition are the quantity p/ (1 - p ), where p is the probability of the event or proposition. The odds against the same event are (1 - p ) /p . In gambling, the odds on display do not represent the true chances that the event will occur, but are the amounts that the bookmaker will pay out on winning bets. For example, suppose that you are willing to make a 1 dollar bet giving 2 to 1 odds that Badgers will win. Then you are willing to pay 2 dollars if Badgers loses in return for receiving 1 dollar if Badgers wins. This means that you think the appropriate probability for Badgers winning is 2/3. Example John and Mary are taking a mathematics course. The course has only three grades: A, B, and C. The probability that John gets a B is .3. The probability that Mary gets a B is .4. The probability that neither gets an A but at least one gets a B is .1.
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Week03Notes - Stat 311 Introduction to Mathematical...

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