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Bayesian point estimation
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11.118. The waiting time to be seated at a restaurant is exponentially distributed with mean 1>u. The prior distribution of u is gamma with mean 0.1 and variance 0.1. A random sample of six customers had an average waiting time of 9 minutes. Find the Bayes estimate for u with (a) squared error (b) absolute error loss function. 11.119. The life-length X of a computer component has the exponential density given by (see page 118) f (x uu) ue ux, x 0 with unknown mean 1>u. Suppose that the prior density of u is gamma with parameters a and b. Based on a random sample of n observations on X, find the Bayes estimate of (a) u and (b) 1>u with respect to the squared error loss function.
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CHAPTER 11 Bayesian Methods
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u) with squared error loss function.
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11.120. In Problem 11.29, find the Bayes estimate of (a) u and (b) u(1
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11.121. In Problem 11.33, find the Bayes estimate of u with squared error loss. 11.122. In Problem 11.26, find the Bayes estimate of u with squared error loss. 11.123. In Problem 11.6, part (a), find the Bayes estimate with squared error loss for the variance of the population, nu(1 u).
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Bayesian interval estimation
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11.124. Ten Bernoulli trials with probability of success u result in five successes. u has the prior density given by 1 p(u) , 0 u 1. Find the 90% Bayes equal tail area credibility interval for u. u(1 u)
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11.125. A random sample of size 10 drawn from a geometric distribution with success probability u yields a mean of 5. The prior density of u is uniform in the interval [0, 1]. Find the 88% equal tail area credibility interval for u. 11.126. In Problem 11.30, find the 85% Bayesian equal tail area credibility interval for u. 11.127. In Problem 11.119, suppose that the prior density of u is gamma with parameters a 0.2 and b 0.15. A random sample of 10 observations on X yielded an average life-length of seven years. Find the 85% equal tail area Bayes credibility interval for (a) u and (b) 1>u.
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Bayesian tests of hypotheses
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11.128. In Problem 11.21, suppose a sample of size 10 yielded the values 2, 0, 1, 1, 3, 0, 2, 4, 2, 2. Test H0 : l against H1 : l 1 using a Bayes 0.05 test. 11.129. In Problem 11.6, assume that n 50 and x 14 and test the null hypothesis H0 : u alternative H1 : u 0.2 using a Bayes 0.025 test. 0.2 against the 1
11.130. Suppose that in Example 11.18 a second sample of 100 observations yielded a mean reaction time of 0.35 sec. Test the null hypothesis H0 : u 0.3 against the alternative H1 : u 0.3 using the Bayes 0.05 test.
Bayes factor
11.131. It is desired to test the null hypothesis u 0.6 against the alternative u 0.6, where u is the probability of success for a Bernoulli trial. Assume that u has a uniform prior distribution on [0, 1] and that in 30 trials there were 17 successes. What is your conclusion if you decide to reject the null hypothesis if BF 1 11.132. The time (in minutes) that a bank customer has to wait in line to be served is exponentially distributed with mean 1>u. The prior distribution of u is gamma with parameters a 0.2 and b 3. A random sample of 10 customers waited an average of 3 minutes. Test the null hypothesis H0 : u 0.7 against H1 : u 0.7 using the Bayes factor rule to reject the null hypothesis if BF 1.
Bayesian predictive distributions
11.133. In Problem 11.13, find the predictive distribution and predictive mean of the number of accidents during the last six months of the year. 11.134. Suppose that 4 successes were obtained in 10 Bernoulli trials with success probability u. An independent set of 5 more Bernoulli trials with the same success probability is being contemplated. Find the predictive distribution of the number of future successes. Assume a prior uniform density for u.
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