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Ratios of Normal Variables and Ratios of Sums of Uniform Variables

George Marsaglia
Journal of the American Statistical Association
Vol. 60, No. 309 (Mar., 1965), pp. 193-204
DOI: 10.2307/2283145
Stable URL: http://www.jstor.org/stable/2283145
Page Count: 12
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Ratios of Normal Variables and Ratios of Sums of Uniform Variables
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Abstract

The principal part of this paper is devoted to the study of the distribution and density functions of the ratio of two normal random variables. It gives several representations of the distribution function in terms of the variate normal distribution of Nicholson's V function, both of which have been extensively studied, and for which tables and computational procedures are readily available. One of these representations leads to an easy derivation of the density function in terms of the Cauchy density and the normal density and integral. A number of graphs of the possible shapes of the density are given, together with an indication of when the density is unimodal or bimodal. The last part of the paper discusses the distribution of the ratio (u1 + ⋯ + un)/(v1 + ⋯ + vm) where the u's and v's are independent, uniform variables. The exact distribution for all n and m is given, and some approximations discussed.

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