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)]).](/images/equations/NormalRatioDistribution/NumberedEquation1.svg) |
(1)
|
From ratio distribution, the distribution of 
is
 |  |  |
(2)
|
 |  | ![1/(2pisigma_xsigma_y)int_(-infty)^infty|y|e^(-[y^2/(2sigma_y^2)+u^2y^2/(2sigma_x^2)])dy](/images/equations/NormalRatioDistribution/Inline13.svg) |
(3)
|
 |  | ![1/(pisigma_xsigma_y)int_0^inftyyexp[-y^2(1/(2sigma_y^2)+(u^2)/(2sigma_x^2))]dy.](/images/equations/NormalRatioDistribution/Inline16.svg) |
(4)
|
But
 |
(5)
|
so
 |  |  |
(6)
|
 |  |  |
(7)
|
 |  |  |
(8)
|
which is a Cauchy distribution.
A more direct derivative proceeds from integration of
 |  |  |
(9)
|
 |  |  |
(10)
|
where 
is a delta function.
