Answer: Suppose that the proportion of defective items in

Consider again the situation described in Example 7.2.8. This time, suppose that the experimenter believes that the prior distribution of is the gamma distribution with parameters 1 and 5000. What would this experimenter compute as the value of Pr(X6 > 3000|x)?

Suppose that the proportion of defective items in a large manufactured lot is known to be either 0.1 or 0.2, and the prior p.f. of is as follows: (0.1) = 0.7 and (0.2) = 0.3. Suppose also that when eight items are selected at random from the lot, it is found that exactly two of them are defective. Determine the posterior p.f. of .

Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which Suppose that the number of defects on a roll of magnetic recording tape has a Poisson distribution for which follows: (1.0) = 0.4 and (1.5) = 0.6. If a roll of tape selected at random is found to have three defects, what is the posterior p.f. of ?

Suppose that the prior distribution of some parameter is a gamma distribution for which the mean is 10 and the variance is 5. Determine the prior p.d.f. of

Suppose that the prior distribution of some parameter is a beta distribution for which the mean is 1/3 and the variance is 1/45. Determine the prior p.d.f. of .

Suppose that the proportion of defective items in a large manufactured lot is unknown, and the prior distribution of is the uniform distribution on the interval [0, 1]. When eight items are selected at random from the lot, it is found that exactly three of them are defective. Determine the posterior distribution of .

Consider again the problem described in Exercise 6, but suppose now that the prior p.d.f. of is as follows: ( ) =  2(1 ) for 0 << 1, 0 otherwise. As in Exercise 6, suppose that in a random sample of eight items exactly three are found to be defectiv

Suppose that X1,...,Xn form a random sample from a distribution for which the p.d.f. is f (x| ), the value of is unknown, and the prior p.d.f. of is ( ). Show that the posterior p.d.f. (|x) is the same regardless of whether it is calculated directly by using Eq. (7.2.7) or s

Consider again the problem described in Exercise 6, and assume the same prior distribution of . Suppose now, however, that instead of selecting a random sample of eight items from the lot, we perform the following experiment: Items from the lot are selected at random one by one until exactly three defectives have been found. If we find that we must select a total of eight items in this experiment, what is the posterior distribution of at the end of the experiment?

Suppose that a single observation X is to be taken from the uniform distribution on the interval [ 1 2 , + 1 2 ], the value of is unknown, and the prior distribution of is the uniform distribution on the interval [10, 20]. If the observed value of X is 12, what is the posterior distribution of ?

Consider again the conditions of Exercise 10, and assume the same prior distribution of . Suppose now, however, that six observations are selected at random from the uniform distribution on the interval [ 1 2 , + 1 2 ], and their values are 11.0, 11.5, 11.7, 11.1, 11.4, and 10.9. Determine the posterior distribution of .