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Linear Creator In Visual Studio .NET Using Barcode creator for VS .NET Control to generate, create 1D image in VS .NET applications. Creating Barcode In Java Using Barcode generator for Java Control to generate, create barcode image in Java applications. 11.60. In the previous problem, assume that u has a uniform prior distribution on [0, 1] and find (a) 95% and (b) 99% equal tail area credibility intervals for u. The posterior distribution of u is beta with parameters 56 and 46 (see Theorem 111). (a) We need the percentiles x0.025 and x0.975 of the preceding beta distribution. These are respectively 0.452 and 0.644. The 95% interval is [0.452, 0.644]. (b) We need the percentiles x0.005 and x0.995 of the preceding beta distribution. These are respectively 0.422 and 0.644. The 99% interval is [0.422, 0.672]. 11.60. In 40 tosses of a coin, 24 heads were obtained. Find a 90% and 99.73% credibility interval for the proportion of heads u that would be obtained in an unlimited number of tosses of the coin. Use a uniform prior for u. By Theorem 111, the posterior density of u is beta with a 25 and b 17. This density has the following percentiles: x0.00135 0.367, x0.05 0.469, x0.95 0.716, x0.99865 0.800. The 90% and 99.73% Bayesian equal tail area credibility intervals are, respectively, [0.469, 0.716] and [0.367, 0.800]. (The traditional confidence intervals are given in Problem 6.15.) 11.62. A sample of 100 measurements of the diameter of a sphere gave a mean x 4.38 inch. Based on prior # experience, we know that the diameter is normally distributed with unknown mean u and variance 0.36. (a) Find 95% and 99% equal tail area credibility intervals for the actual diameter u assuming a normal prior density with mean 4.5 inches and variance 0.4. (b) With what degree of credibility could we say that the true diameter is 4.38 0.01 (a) From Theorem 113, we see that the posterior mean and variance for u are 4.381 and 0.004. The 95% credibility interval is [4.381 (1.96 0.063), 4.381 (1.96 0.063)] [4.26, 4.50]. Similarly, the 90% credibility interval is [4.381 (1.645 0.063), 4.381 (1.645 0.063)] [4.28, 4.48]. (b) We need the area under the posterior density from 4.37 to 4.39. This equals the area under the standard normal density between (4.37 4.381)>0.063 0.17 and (4.39 4.381)>0.063 0.14. This equals 0.1232, so the required degree of credibility is roughly 12%.

