The Distribution of the Ratio of Jointly Normal Variables

We derive the probability density of the ratio of components of the bivariate normal distribution with arbitrary parameters. The density is a product of two factors, the first is a Cauchy density, the second a very complicated function. We show that the distribution under study does not possess an expected value or other moments of higher order. Our particular interest is focused on the shape of the density. We introduce a shape parameter and show that according to its sign the densities are classified into three main groups. As an example, we derive the distribution of the ratio ) /( 1 m m mB B Z − − = for a polynomial regression of order m. For 1 = m , Z is the estimator for the zero of a linear regression, for 2 = m , an estimator for the abscissa of the extreme of a quadratic regression, and for 3 = m , an estimator for the abscissa of the inflection point of a cubic regression.