Tuesday, August 25, 2015

The Distribution of a Ratio of Correlated Normals

Suppose that the random variables X1 and X2 are jointly distributed as bivariate Normal, with means of θ1 and θ2, variances of σ12 and σ22 respectively, and a correlation coefficient of ρ.

In this post we're going to be looking at the distribution of the ratio, W = (X1 / X2).

You probably know that if X1 and X2 are independent standard normal variables, then W follows a Cauchy distribution. This will emerge as a special case in what follows.

The more general case that we're concerned with is of interest to econometricians for several reasons.