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).
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