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).

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1112 m F