Normal Ratio Distribution
GaussianRatioDistribution
The ratio X/Y of independent normally distributed variates with zero mean is distributed with a Cauchy distribution. This can be seen as follows. Let X and Y both have mean 0 and standard deviations of sigma_x and sigma_y, respectively, then the joint probability density function is the bivariate normal distribution with rho=0,
![画像: f(x,y)=1/(2pisigma_xsigma_y)e^(-[x^2/(2sigma_x^2)+y^2/(2sigma_y^2)]).](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/NormalRatioDistribution/NumberedEquation1.svg){kind=link}
From ratio distribution, the distribution of U=X/Y is
P(u) = [画像:int_(-infty)^infty|y|f(uy,y)dy]
(2)
![画像:1/(2pisigma_xsigma_y)int_(-infty)^infty|y|e^(-[y^2/(2sigma_y^2)+u^2y^2/(2sigma_x^2)])dy](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/NormalRatioDistribution/Inline13.svg){kind=link}
![画像:1/(pisigma_xsigma_y)int_0^inftyyexp[-y^2(1/(2sigma_y^2)+(u^2)/(2sigma_x^2))]dy.](/index.cgi/larger-text/https://mathworld.wolfram.com/images/equations/NormalRatioDistribution/Inline16.svg){kind=link}
But
so
which is a Cauchy distribution.
A more direct derivative proceeds from integration of
where delta(x) is a delta function.
See also
Cauchy Distribution, Normal Difference Distribution, Normal Distribution, Normal Product Distribution, Normal Sum DistributionExplore with Wolfram|Alpha
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Cite this as:
Weisstein, Eric W. "Normal Ratio Distribution." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/NormalRatioDistribution.html