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 ,
(1)

From ratio distribution, the distribution of is
(2)
 
(3)
 
(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.