Show that two random variables and cannot have the joint

Suppose that a random variable X has the normal distribution with mean and precision . Show that the random variable Y = aX + b (a = 0) has the normal distribution with mean a + b and precision /a2

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean ( < < ) and known precision . Suppose also that the prior distribution of is the normal distribution with mean 0 and precision 0. Show that the posterior distribution of, given that Xi = xi (i = 1, . . . , n) is the normal distribution with mean 00 + nxn 0 + n and precision 0 + n .

Suppose that X1,...,Xn form a random sample from the normal distribution with known mean and unknown precision ( > 0). Suppose also that the prior distribution of is the gamma distribution with parameters 0 and 0 (0 > 0 and 0 > 0). Show that the posterior distribution of given that Xi = xi (i = 1, . . . , n) is the gamma distribution with parameters 0 + (n/2) and 0 + 1 2  N i=1 (xi

Suppose that X1,...,Xn are i.i.d. having the normal distribution with mean and precision given (, ). Let (, ) have the usual improper prior. Let 2 = s2 n/(n 1). Prove that the posterior distribution of V = (n 1)2 is the 2 distribution with n 1 degrees of freedom.

Suppose that two random variables and have the joint normal-gamma distribution such that E() = 5. Var() = 1, E( ) = 1/2, and Var( ) = 1/8. Find the prior hyperparameters 0, 0, 0, and 0 that specify the normal-gamma distribution.

Show that two random variables and cannot have a joint normal-gamma distribution such that E() = 0, Var() = 1, E( ) = 1/2, and Var( ) = 1/4

Show that two random variables and cannot have the joint normal-gamma distribution such that E() = 0, E( ) = 1, and Var( ) = 4.

Suppose that two random variables and have the joint normal-gamma distribution with hyperparameters 0 = 4, 0 = 0.5, 0 = 1, and 0 = 8. Find the values of (a) Pr( > 0) and (b) Pr(0.736 << 15.680).

Using the prior and data in the numerical example on nursing homes in New Mexico in this section, find (a) the shortest possible interval such that the posterior probability that lies in the interval is 0.90, and (b) the shortest possible confidence interval for for which the confidence coefficient is 0.90.

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and unknown precision , and also that the joint prior distribution ofand is the normal-gamma distribution satisfying the following conditions: E() = 0, E( ) = 2, E( 2) = 5, and Pr(|| < 1.412) = 0.5. Determine the prior hyperparameters 0, 0, 0, and 0.

Consider again the conditions of Exercise 10. Suppose also that in a random sample of size n = 10, it is found that xn = 1 and s2 n = 8. Find the shortest possible interval such that the posterior probability that lies in the interval is 0.95.

Suppose that X1,...,Xn form a random sample from the normal distribution with unknown mean and unknown precision , and also that the joint prior distribution ofand is the normal-gamma distribution satisfying the following conditions: E( ) = 1, Var( ) = 1/3, Pr( > 3) = 0.5, and Pr( > 0.12) = 0.9. Determine the prior hyperparameters 0, 0, 0, and 0.

Consider again the conditions of Exercise 12. Suppose  also that in a random sample of size n = 8, it is found that n i=1 xi = 16 and n i=1 x2 1 = 48. Find the shortest possible interval such that the posterior probability that lies in the interval is 0.99.

Continue the analysis in Example 8.6.2 on page 498. Compute an interval (a, b) such that the posterior probability is 0.9 that a<

We will draw a sample of size n = 11 from the normal distribution with mean and precision . We will use a natural conjugate prior for the parameters (, ) from the normal-gamma family with hyperparameters 0 = 2, 0 = 1, 0 = 3.5, and 0 = 2. The sample yields an average of xn = 7.2 and s2 n = 20.3. a. Find the posterior hyperparameters. b. Find an interval that contains 95% of the posterior distribution of .

The study on acid concentration in cheese included a total of 30 lactic acid measurements, the 10 given in Example 8.5.4 on page 487 and the following additional 20: 1.68, 1.9, 1.06, 1.3, 1.52, 1.74, 1.16, 1.49, 1.63, 1.99, 1.15, 1.33, 1.44, 2.01, 1.31, 1.46, 1.72, 1.25, 1.08, 1.25. a. Using the same prior as in Example 8.6.2 on page 498, compute the posterior distribution of and based on all 30 observations. b. Use the posterior distribution found in Example 8.6.2 on page 498 as if it were the prior distribution before observing the 20 observations listed in this problem. Use these 20 new observations to find the posterior distribution of and and compare the result to the answer to part (a). 17. Consider the analysi

Consider the analysis performed in Example 8.6.2. This time, use the usual improper prior to compute the posterior distribution of the parameters

Treat the posterior distribution conditional on the first 10 observations found in Exercise 17 as a prior and then observe the 20 additional observations in Exercise 16. Find the posterior distribution of the parameters after observing all of the data and compare it to the distribution found in part (b) of Exercise 16.

Consider the situation described in Exercise 7 of Sec. 8.5. Use a prior distribution from the normal-gamma family with values 0 = 1, 0 = 4, 0 = 150, and 0 = 0.5. a. Find the posterior distribution of and = 1/2. b. Find an interval (a, b) such that the posterior probability is 0.90 that a<

Consider the calorie count data described in Example 7.3.10 on page 400. Now assume that each observation has the normal distribution with unknown mean and unknown precision given the parameter (, ). Use the normal-gamma conjugate prior distribution with prior hyperparameters 0 = 0, 0 = 1, 0 = 1, and 0 = 60. The value of s2 n is 2102.9. a. Find the posterior distribution of (, ). b. Compute Pr( > 1|x).