Ratio Distribution - Gaussian Ratio Distribution

Gaussian Ratio Distribution

When X and Y are independent and have a Gaussian distribution with zero mean the form of their ratio distribution is fairly simple: It is a Cauchy distribution. However, when the two distributions have non-zero means then the form for the distribution of the ratio is much more complicated. In 1969 David Hinkley found a form for this distribution. In the absence of correlation (cor(X,Y) = 0), the probability density function of the two normal variable X = N(μ_X, σ_X2) and Y = N(μ_Y, σ_Y2) ratio Z = X/Y is given by the following expression:

where

The above expression becomes even more complicated if the variables X and Y are correlated. It can also be shown that p(z) is a standard Cauchy distribution if μ_X = μ_Y = 0, and σ_X = σ_Y = 1. In such case b(z) = 0, and

If, or the more general Cauchy distribution is obtained

where ρ is the correlation coefficient between X and Y and

The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.