Normal Ratio Distribution
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12-4-2021
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Normal Ratio Distribution
The ratio 
of independent normally distributed variates with zero mean is distributed with a Cauchy distribution. This can be seen as follows. Let 
and 
both have mean 0 and standard deviations of 
and 
, respectively, then the joint probability density function is the bivariate normal distribution with 
,
![f(x,y)=1/(2pisigma_xsigma_y)e^(-[x^2/(2sigma_x^2)+y^2/(2sigma_y^2)]).](https://mathworld.wolfram.com/images/equations/NormalRatioDistribution/NumberedEquation1.gif) |
(1)
|
From ratio distribution, the distribution of 
is
But
 |
(5)
|
so
which is a Cauchy distribution.
A more direct derivative proceeds from integration of
where 
is a delta function.
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