Distribution of the Ratio of Quadratic Forms in Normal Variables—Numerical Methods

This paper is concerned mostly with the distribution of the ratio of two quadratic forms in normal variables. The probability density of the ratio is obtained as an integral and closed-form expressions are given for several special cases. However, in the general case numerical methods must be used to evaluate the density. A method of numerical integration is presented and illustrated by an example. The behavior of the density around the points at which it is irregular is examined. The general case of the distribution of the ratio of two composite random variables is discussed briefly.