Statistical Papers

, Volume 54, Issue 2, pp 309–323 | Cite as

On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables

  • Eloísa Díaz-FrancésEmail author
  • Francisco J. Rubio
Regular Article


The distribution of the ratio of two independent normal random variables X and Y is heavy tailed and has no moments. The shape of its density can be unimodal, bimodal, symmetric, asymmetric, and/or even similar to a normal distribution close to its mode. To our knowledge, conditions for a reasonable normal approximation to the distribution of ZX/Y have been presented in scientific literature only through simulations and empirical results. A proof of the existence of a proposed normal approximation to the distribution of Z, in an interval I centered at βE(X) /E(Y), is given here for the case where both X and Y are independent, have positive means, and their coefficients of variation fulfill some conditions. In addition, a graphical informative way of assessing the closeness of the distribution of a particular ratio X/Y to the proposed normal approximation is suggested by means of a receiver operating characteristic (ROC) curve.


Coefficient of variation Ratio of normal means ROC curve 

Mathematics Subject Classification (2000)



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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Centro de Investigación en Matemáticas (CIMAT)GuanajuatoMexico
  2. 2.Department of StatisticsUniversity of WarwickCoventryUK

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