Section 6.3 (p.327)

4. By Theorem 2, the posterior distribution is a Gamma distribution with parameters a = 3+S x_i = 16 and b= 1+n = 6.

8. By Theorem 3, the prior distribution can also be a normal distribution with mean m and variance n², and

Solve the above equations for m and n², we get m = 0 and n² = 1/5.

14. (a) Let y = 1/q. Then

(b) Let X₁, …, X_n be a sample from Normal(m, q). Then

Therefore,

which has the same form as x(q) with new a = a + n/2 and new b = b + ½ S (x_i-m)².

16. We have

Now max(x₁, …, x₃) = 8, therefore

It follows that

18.

This is a Gamma distribution with parameters a₁ = n+a and b₁ = b-S log x_i. From sec. 5.9, the posterior mean is a₁/b₁ and the posterior variance is .