In this exercise you will prove Theorem 9.8.2. a. Prove that the joint p.d.f. of the data given the parameters 1, 2, and can

In this exercise you will prove Theorem 9.8.2.
a. Prove that the joint p.d.f. of the data given the parameters μ1, μ2, and τ can be written as a constant times

b. Multiply the prior p.d.f. times the p.d.f. in part (a). Bayes€™ theorem for random variables says that the result is proportional (as a function of the parameters) to the posterior p.d.f.
i. Show that the posterior p.d.f., as a function of μ1 for fixed μ2 and Ï„ , is the p.d.f. of the normal distribution with mean m and variance (mÏ„)ˆ’1.
ii. Show that the posterior p.d.f., as a function of μ2 for fixed μ1 and Ï„ , is the p.d.f. of the normal distribution with mean n and variance (nÏ„ )ˆ’1.
iii. Show that, conditional on τ , μ1 and μ2 are independent with the two normal distributions found above.
iv. Show that the marginal posterior distribution of Ï„ is the gamma distribution with parameters (m + n ˆ’ 2)/2 and (s2x + s2y)/2.
c. Show that the conditional distribution of

given Ï„ is a standard normal distribution and hence Z is independent of Ï„ .
d. Show that the distribution of W = (s2x + s2y)Ï„ is the gamma distribution with parameters (m + n ˆ’ 2)/2 and 1/2, which is the same as the χ2 distribution with m + n ˆ’ 2 degrees of freedom.
e. Prove that Z/(W/(m + n ˆ’ 2))1/2 has the t distribution with m + n ˆ’ 2 degrees of freedom and that it equals the expression in Eq. (9.8.17).

1112 m F