<- file stat 97ratio.html ->

Ratio of two normals

=======================Dick Startz, 02 May 1997==========ssc From: startz@u.washington.edu (Dick Startz) Subject: Re: Ratio of Indepedent Normals Message-ID: <336a3b2d.6283370@news.u.washington.edu> If it helps, the exact distribution of the ratio of two normal variables for arbitrary mean, variance, and correlation is a well-solved problem - albeit one with an ugly answer. See Hinckley, D. V. (1969), "On the Ratio of Two Correlated Normal Random Variables," Biometrika, 56, 635-639. who cites as an earlier source Fieller, E.C. (1932), "The distribution of the index in a normal bivariate population," Biometrika 24, 428-40. -Dick Startz University of Washington On Fri, 2 May 1997 09:53:43 -0500, Chris Ward <chrisw@MAILHOST.TRDLNK.COM> wrote: >Subject Ratio of Two Normally Distributed Random Variables > > >I trust we all accept that that ratio of 2 N(0,1) independent >random variables is a Cauchy Distribtuion with undefined mean >and variance and that neither a Taylor Expansion or any means >of simulation will make this most intractable distribution easier >to deal with. > >However that ratio of a N(a,1)/N(b,1) independent random variables >certainly approaches normal as |a|>>1 and |b|>>1 , in in this >instance the Taylor expansion gives a good approximation to the >resulting mean and standard devation. > >(generate a set with a=b=10) > >mean ~ (a/b) >stdev ~ ( 1/a^2 + 1/b^2 ) > >Which is the standard results for the mean and variance of a the ratio >of two independant measurements with (approximately) normally distributed >errors. > >What remains is the form of the limiting process as |a| and |b| ->0. > > >Can someone tackle this problem and end the flood of caustic emails ? =======================George Marsaglia, 8 July 2001==========ssm < snip. On subject 1/X, X normal > That 97ratio.html states: If it helps, the exact distribution of the ratio of two normal variables for arbitrary mean, variance, and correlation is a well-solved problem - albeit one with an ugly answer. See Hinckley, D. V. (1969), "On the Ratio of Two Correlated Normal Random Variables," Biometrika, 56, 635-639. In that article, Hinkley seemed unaware that a linear transformation will take the ratio of correlated normals into a ratio of independent ones, even though the matter was thoroughly covered in Marsaglia,G. (1965),``Ratios of normal variables and ratios of sums of uniform variables", J. Amer. Statis. Assoc. 60, 193--204, which was pointed out to, but ignored by, the editors of Biometrika. The latter article is available through JSTOR. It describes the wide variety of densities that can arise from the ratio of normals, methods for evaluating them and ways to determine which are unimodal. George Marsaglia =======================Robert Dodier, 07 July 2001==========ssm < case of 1/X > For this special case, the solution is quite simple. In general, if y=g(x) is an invertible transformation of the continuous variable x, then the density of y is just q(y) = p( g^{-1}(y) ) | g^{-1}'(y) |, writing p for the density of x and q for that of y. In this case we have y=g(x)=1/x, g^{-1}(y) = 1/y, omitting x=y=0. Then g^{-1}'(y) = -1/y^2, so q(y) = p(1/y)/y^2. The problem mentioned above states that p is a normal density, p(x)=exp(-(1/2) ((x-mu)/sigma)^2)/(sigma sqrt{2 pi}). However, the formula for q is the same whatever p is; one might very reasonably choose p to be a truncated normal, to avoid the region around 0. As pointed out by other posters, a general solution for the ratio of normal variables does exist, and it is messy. http://www.seanet.com/~ksbrown has a nice discussion under the heading "Probability and Statistics". subheading "Ratio Populations". [July, 2002 note: working address is http://mathpages.com/home/kmath042/kmath042.htm ] * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

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