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

TL;DR

This paper derives the exact distribution of the ratio of products of independent zero-mean normal variables, providing formulas for density, distribution, characteristic functions, and asymptotic properties.

Contribution

It introduces the exact probability density function for the ratio of products of independent normal variables, a novel analytical result.

Findings

Exact density function derived for the ratio of products

Formulas for cumulative distribution and characteristic functions

Asymptotic approximations for tail probabilities and quantiles

Abstract

Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $\sigma_{X_i}^2$, $i=1,\ldots,M$, and $\sigma_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$. 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.