BDA3 Chapter 3 Exercise 10

Here’s my solution to exercise 10, chapter 3, of Gelman’s Bayesian Data Analysis (BDA), 3rd edition. There are solutions to some of the exercises on the book’s webpage.

For j=1,2, let

yj∣μjσ2j∼Normal(μj,σ2j)p(μj,logσ2j)∝1.

We show that

s21σ22s22σ21∼F(n1−1,n2−1).

Equation 3.5 in the book shows that σ2j∣y∼InvChi2(nj−1,s2j). It follows that σ2j(nj−1)s2j∼\dinvChi(nj−1). Thus, (nj−1)s2jσ2j∼Chi2(nj−1). The result follows from the fact that the ratio of two χ2 random variables (divided by the ratio of their degrees of freedom) has an F-distribution.