Advanced Distribution Theory

Studying distributions of functions of several random variables is of primary interest in probability and statistics. For example, the original variables X 1, X 2, …, X n could be the inputs into some process or system, and we may be interested in the output, which is some suitable function of these input variables. Sums, products, and quotients are special functions that arise quite naturally in applications. These are discussed with a special emphasis in this chapter, although the general theory is also presented. Specifically, we present the classic theory of polar transformations and the Helmert transformation in arbitrary dimensions, and the development of the Dirichlet, t- and the F-distribution. The t- and the F-distribution arise in numerous problems in statistics, and the Dirichlet distribution has acquired an extremely special role in modeling and also in Bayesian statistics. In addition, these techniques and results are among the most sophisticated parts of distribution theory.