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On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables. (English) Zbl 1364.62039
Summary: 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 $$Z=X/Y$$ 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 $$\beta=E(X)/E(Y)$$, 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.

##### MSC:
 62E17 Approximations to statistical distributions (nonasymptotic) 62P10 Applications of statistics to biology and medical sciences; meta analysis
##### Keywords:
coefficient of variation; ratio of normal means; ROC curve
zoverw
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##### References:
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