3. Consider again the clinical trial design setting of O’Hagan and Stevens

(2001), as described in Subsection 6.2.2.

(a) Confirm that expression (6.11) is the result of the analysis objective P(β > 0|y) > ω for this model. (Hint: Use expression (4.2), the Lindley and Smith (1972) result on the normality of the posterior in normal linear models with known variances, as well as standard formulae for linear transformations of multivariate normal distributions.)

(b) Confirm that expression (6.12) is the result of the design objective

(6.10) for this model. (Hint: This time use expression (4.1), the Lindley and Smith (1972) result on the normality of the marginal distribution of the data in normal linear models with known variances.)

(c) Confirm that expression (6.14) is the explicit solution to the the frequentist version of this Bayesian sample size problem in the above setting when we insist n = n1 = n2.