On the Joint Distribution of a Quadratic and a Linear Form in Normal Variables

In this paper a series representation of the joint density and the joint distribution of a quadratic form and a linear form in normal variables is developed. The expansion makes use of Laguerre polynomials. As an example the calculation of the joint distribution of the mean and the sample variance is considered. The truncated series is compared with the empirical distribution function which was determined in a Monte Carlo study.