Ratios of Normal Variables and Ratios of Sums of Uniform Variables

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 vivariate normal distribution and 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 (u ₁+ ¨˙ +u_n )/(v ₁+ ¨˙ +v_m ) 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.