Density of the Ratio of Two Normal Random Variables and Applications

Abstract

In reply to a question raised in the literature, and to settle an argument debated in the last decades, we give the exact closed form expression of the density of X/Y, where X and Y are normal random variables, in terms of Hermite and confluent hypergeometric functions. All cases will be considered: standardized and nonstandardized variables, independent or correlated variables. Examples in applied disciplines are presented, and generalizations to ratios of variables from scale mixtures of bivariate normal distributions show the potential of further new applications in applied statistics and operations research.

Keywords:

• Bivariate normal
• Finite sampling
• Hermite function
• Integral representation
• Kummer confluent hypergeometric function
• Normal
• Ratio

AMS Classification:

• 62E15
• 62N05

Acknowledgments

The research of Pham-Gia and Marchand partially supported by NSERC of Canada. The authors wish to thank a referee for some pertinent comments that have helped to improve the presentation of the paper.