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Chapter 7: Q2E (page 425)
Question:It is not known what proportion p of the purchases of a certain brand of breakfast cereal are made by women and what proportion are made by men. In a random sample of 70 purchases of this cereal, it was found that 58 were made by women and 12 were made by men. Find the M.L.E. of p.
0.828
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Let 胃denote the average number of defects per 100 feet of a certain type of magnetic tape. Suppose that thevalue of胃is unknown and that the prior distribution of胃is the gamma distribution with parameters伪=2 and
尾=10. When a 1200-foot roll of this tape is inspected,exactly four defects are found. Determine the posteriordistribution of胃.
Consider the data in Example 7.3.10. This time, suppose that we use the improper prior 鈥減.d.f.鈥漒(\xi \left( \theta \right) = 1\)(for all 胃). Find the posterior distribution of\(\theta \)and the posterior probability that\(\theta > 1\).
Suppose that a random sample of size n is taken from the Bernoulli distribution with parameter 胃, which is unknown, and that the prior distribution of 胃 is a beta distribution for which the mean is\({\mu _0}\). Show that the mean of the posterior distribution of 胃 will be a weighted average having the form \({\gamma _n}{\overline X _n} + \left( {1 - {\gamma _n}} \right){\mu _0}\)and show that \({\gamma _n} \to 1\)as\(n \to \infty \).
Suppose that a random sample is to be taken from a normal distribution for which the value of the mean 胃 is unknown and the standard deviation is 2, the prior distribution of 胃 is a normal distribution for which the standard deviation is 1, and the value of 胃 must be estimated by using the squared error loss function. What is the smallest random sample that must be taken in order for the mean squared error of the Bayes estimator of 胃 to be 0.01 or less? (See Exercise 10 of Sec. 7.3.)
Question: Suppose that the lifetime of a certain type of lamp has an exponential distribution for which the value of the parameter \({\bf{\beta }}\) is unknown. A random sample of n lamps of this type are tested for a period of T hours and the number X of lamps that fail during this period is observed, but the times at which the failures occurred are not noted. Determine the M.L.E. of \({\bf{\beta }}\) based on the observed value of X.
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