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 ,

(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.