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Sampling from a binomial distribution
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11.106. The number of defective tools in each lot of 10 produced by a manufacturing process has a binomial distribution with parameter u. Assume a vague prior density for u (uniform on (0, 1)) and determine its posterior density based on the information that two defective tools were found in the last lot that was inspected. 11.107. In 50 tosses of a coin, 32 heads were obtained. Find the posterior distribution of the proportion of heads u that would be obtained in an unlimited number of tosses of the coin. Use a noninformative prior (uniform on (0, 1)) for the unknown probability.
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CHAPTER 11 Bayesian Methods
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11.108. Continuing the previous problem, suppose an additional 50 tosses of the coin were made and 35 heads were obtained. Find the latest posterior density.
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Sampling from a Poisson distribution
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11.109. The number of accidents during a six-month period at an intersection has a Poisson distribution with mean l. It is believed that l has a gamma prior density with parameters a 2 and b 5. If a total of 14 accidents were observed during the first six months of the year, find the (a) posterior density, (b) posterior mean, and (c) posterior variance. 11.110. The number of defects in a 2000-foot spool of yarn manufactured by a machine has a Poisson distribution with unknown mean l. The prior distribution of l is gamma with parameters a 4 and b 2. A total of 42 defects were found in a sample of 10 spools that were examined. Determine the posterior density of l.
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11.111. A random sample of 16 observations is taken from a normal population with unknown mean u and variance 9. The prior distribution of u is standard normal. Find (a) the posterior mean, (b) its precision, and (c) the precision of the maximum likelihood estimator. 11.112. The reaction time of an individual to certain stimuli is known to be normally distributed with unknown mean u but a known standard deviation of 0.30 sec. A sample of 20 observations yielded a mean reaction time of 2 sec. Assume that the prior density of u is normal with mean 1.5 sec. and variance y2 0.10. Find the posterior density of u.
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11.113. The random variable X has the Poisson distribution with parameter l. The prior distribution of l is given p(l) 1> !l, l 0. A random sample of 10 observations on X yielded a sample mean of 3.5. Find the posterior density of l. 11.114. A population is known to be normal with mean 0 and unknown variance u. The variance has the improper prior density p(u) 1> !u, u 0. If a random sample of size 5 from the population consists of 2.5, 3.2, 1.8, 2.1, 3.1, find the posterior distribution of u.
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Conjugate prior distributions
11.115. A random sample of size 20 drawn from a geometric distribution with parameter u (see page 117) yields a mean of 5. The prior density of u is uniform in the interval [0, 1]. Determine the posterior distribution of u. 11.116. The interarrival time of customers at a bank is exponentially distributed with mean 1>u, where u has a gamma distribution with parameters a 1 and b 2. Ten customers were observed over a period of time and were found to have an average interarrival time of 5 minutes. Find the posterior distribution of u. 11.117. A population is known to be normal with mean 0 and unknown variance u. The variance has the inverse gamma prior density with parameters a 1 and b 1 (see Problem 11.99). Find the posterior distribution of u based on the following random sample from the population: 2, 1.5, 2.5, 1.