Distribution Function for the ratio of Two Normal Random Variables

The distribution of the ratio of two normal random variables X and Y was studied from [1] (the density function) and [2] (the distribution function). The shape of its density function can be uni-modal, bimodal, symmetric, asymmetric, following several type of distributions, like Dirac Distribution, Normal Distribution, Cauchy Distribution or Recinormal Distribution. In this paper we study a different approximation for this distribution Z = X/Y, as a function of four parameters: ratio of the means of the two normal variables, ratio of the standard deviations of the two normal variables, the variation coefficient of the normal variable Y, and the correlation between the two variables. A formula for the Distribution function and the density function of Z is given. In addition, using graphical procedures we established singularity points for the parameters where the approximation given for Z has a non normal shape.