Distributions of ratios of random variables from the power-quadratic exponential family and applications

Abstract

We derive the closed form exact expression of the density of the ratio X ₁/X ₂, where X ₁ and X ₂ are independent and belong to the very versatile family of power-quadratic exponential distributions. The case of X ₂ as a gamma variable is studied in detail, and new applications found in different domains show the varied uses of the results presented.

Keywords:

• Power-quadratic exponential
• Gamma
• Ratio of random variables
• Hermite function
• Tricomi function
• Rayleigh
• Beta type 2
• Student-type

Acknowledgement

This research was partially supported by NSERC Grant A9249 (Canada). The authors wish to thank Professor Jeannette Dolorese LeBlanc for very helpful discussions on this topic and an anonymous referee for suggesting an interesting general result, which is presented as theorem, as well as offering other useful comments.