Published by the Foundation for Open Access Statistics Editors-in-chief: Bettina Grün, Torsten Hothorn, Rebecca Killick, Edzer Pebesma, Achim Zeileis    ISSN 1548-7660; CODEN JSSOBK
Authors: George Marsaglia
Title: Ratios of Normal Variables
Abstract: This article extends and amplifies on results from a paper of over forty years ago. It provides software for evaluating the density and distribution functions of the ratio z/w for any two jointly normal variates z,w, and provides details on methods for transforming a general ratio z/w into a standard form, (a+x)/(b+y) , with x and y independent standard normal and a, b non-negative constants. It discusses handling general ratios when, in theory, none of the moments exist yet practical considerations suggest there should be approximations whose adequacy can be verified by means of the included software. These approximations show that many of the ratios of normal variates encountered in practice can themselves be taken as normally distributed. A practical rule is developed: If a < 2.256 and 4 < b then the ratio (a+x)/(b+y) is itself approximately normally distributed with mean μ = a/(1.01b - .2713) and variance σ2 = (a2 + 1)/(b2 + .108b - 3.795) μ2.

Page views:: 19252. Submitted: 2006-03-07. Published: 2006-05-11.
Paper: Ratios of Normal Variables     Download PDF (Downloads: 31986)
Supplements: zoverw.c: Sample C routines Download (Downloads: 1544; 2KB)

DOI: 10.18637/jss.v016.i04

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Paper: Creative Commons Attribution 3.0 Unported License
Code: GNU General Public License (at least one of version 2 or version 3) or a GPL-compatible license.