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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

D(u)=P(U<=u)
(1)
=P(Y<=uX|X>0)+P(Y>=uX|X<0)
(2)
=int_0^inftyint_0^(ux)f(x,y)dydx+int_(-infty)^0int_(ux)^0f(x,y)dydx.
(3)

The probability function is then

P(u)=D^'(u)
(4)
=int_0^inftyxf(x,ux)dx-int_(-infty)^0xf(x,ux)dx
(5)
=int_(-infty)^infty|x|f(x,ux)dx.
(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,
(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.
(8)

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. https://mathworld.wolfram.com/RatioDistribution.html

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