Ratio Distribution

Given two distributions Y and X with joint probability density function f(x,y), let U=Y/X be the ratio distribution. Then the distribution function of u is


The probability function is then


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,
 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.

See also

Cauchy Distribution, Difference Distribution, Product Distribution, Sum Distribution, Uniform Ratio Distribution

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Cite this as:

Weisstein, Eric W. "Ratio Distribution." From MathWorld--A Wolfram Web Resource.

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