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CHAPTER 11 Bayesian Methods
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EXAMPLE 11.26 The shipping weight of packages handled by a company is normally distributed with mean u lb and variance 8. If the first 25 packages handled on a given day have an average weight of 15 lb what are the chances the next 25 packages to be handled will have an average in excess of 16 lb Assume that u has a prior normal distribution with mean 12 and variance 9. From Theorem 11-12, the mean and variance of the predictive density of the average weight of the future sample are # given by 14.90 and 0.31. The probability we need is P(Y 16) P(Z 1.98) 0.0234, so the chances are about 2% that the future average weight would exceed 16 lbs.
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Point and interval summaries of the predictive density may be obtained as for the posterior density of a parameter, and they serve similar purposes. For example, given the predictive density function f *(y) of the sample # # mean Y of a future sample from a population, the expectation, median, or mode of f *(y) may be used as a pre# # dictive point estimate of Y. Also, intervals [yL, yU] satisfying the property
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yU * # # 3 f (y) dy yL
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# may be used as Bayesian (1 a) 100% predictive intervals for Y; and equal tail area and HPD predictive intervals may be defined as in the case of credibility intervals for a parameter.
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EXAMPLE 11.27 In Example 11.24, find the predictive (a) mean, (b) median, and (c) mode of the number of future successes.
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(a) The predictive distribution of Y is given in Table 11-5. The predictive mean number of future successes is the expectation of Y, which is 5.34. (b) The predictive median is between 5 and 6, and we may take it to be 5.5. (c) The predictive mode is 6.
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EXAMPLE 11.28 In Example 11.26, find a 95% equal tail area predictive interval for the average weight of the 25 future packages. The predictive distribution is normal with mean 14.90 and variance 0.31. The 95% equal tail interval is given by 14.90 (1.96 0.56) [13.8, 16.0].
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Subjective probability 11.1. Identify the type of probability used: (a) The probability that my daughter will attend college is 0.9. (b) The chances of getting three heads out of three tosses of a fair coin are 1 in 8. (c) I am 40% sure it will rain on the 4th of July this year because it did in 12 out of the past 30 years. (d) We are 70% sure that the variance of this distribution does not exceed 3.5. (e) Some economists believe there is a better than even chance that the economy will go into recession next year. (f) The chances are just 2% that she will miss both of her free throws. (g) I am 90% sure this coin is not fair. (h) The probability that all three children are boys in a three-child family is about 0.11. (i) The odds are 3 to 1 the Badgers will not make it to the Super Bowl this year. ( j) You have one chance in a million of winning this lottery. (k) You have a better than even chance of finding a store that carries this item.
(a), (d), (e), (g), (i): subjective; (b), ( j), (k): classical; (c), (f), (h): frequency
Prior and posterior distributions 11.2. A box contains two fair coins and two biased coins (each with P( heads ) 0.3). A coin is chosen at random from the box and tossed four times. If two heads and two tails are obtained, find the posterior probabilities of the event F that the chosen coin is fair and the event B that the coin is biased.