The distribution of the ratio of products of independent zero mean normal random variables

Let X₁,… ,XM X 1 , … , X M and Y₁,… ,YN Y 1 , … , Y N be independent zero mean normal random variables with variances σ Xᵢ² σ X i 2 , i=1,… ,M i = 1 , … , M , and σ Yⱼ² σ Y j 2 , j=1,… ,N j = 1 , … , N , respectively, and let X=X₁⋯ XM X = X 1 ⋯ X M and Y=Y₁⋯ YN Y = Y 1 ⋯ Y N . In this paper, we derive the exact probability density function of the ratio X / Y . We apply this formula to derive exact formulas for the cumulative distribution function and the characteristic function. We also obtain further distributional properties, including asymptotic approximations for the probability density function, tail probabilities and the quantile function.