Given two distributions 
and 
with joint probability density function 
, let 
be the ratio distribution. Then the distribution function
of 
is
 |  |  |
(1)
|
 |  |  |
(2)
|
 |  |  |
(3)
|
The probability function is then
 |  |  |
(4)
|
 |  |  |
(5)
|
 |  |  |
(6)
|
For variates with standard normal distributions, the ratio distribution is a Cauchy distribution.
For a uniform ratio distribution
![f(x,y)={1 for x,y in [0,1]; 0 otherwise,](/images/equations/RatioDistribution/NumberedEquation1.svg) |
(7)
|
![P(u)={0 u<0; int_0^1xdx=[1/2x^2]_0^1=1/2 for 0<=u<=1; int_0^(1/u)xdx=[1/2x^2]_0^(1/u)=1/(2u^2) for u>1.](/images/equations/RatioDistribution/NumberedEquation2.svg) |
(8)
|
