Bayesian Statistics: An Introduction, 4th Edition

A.9 Normal/chi-squared distribution

The ordered pair (X, Y) has a normal/chi-squared distribution if

for some S and ν and, conditional on Y,

for some μ and n. An equivalent condition is that the joint density function (for and ) is

where

If we define

then the marginal distribution of X is given by the fact that

The marginal distribution of Y is of course

Approximate methods of constructing two-dimensional highest density regions for this distribution are described in Box and Tiao (1992, Section 2.4).