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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

$\displaystyle D(u)$ $\textstyle =$ $\displaystyle P(U\leq u)$  
  $\textstyle =$ $\displaystyle P(Y\leq uX \vert X>0)+P(Y\geq uX\vert X<0)$  
  $\textstyle =$ $\displaystyle \int_0^\infty \int_0^{ux} f(x,y)\,dy\,dx+\int_{-\infty}^0\int_{ux}^0 f(x,y)\,dy\,dx.$  

The probability function is then
$\displaystyle P(u)$ $\textstyle =$ $\displaystyle D'(u)=\int_0^\infty xf(x,ux)\,dx-\int_{-\infty}^0 xf(x,ux)\,dx$  
  $\textstyle =$ $\displaystyle \int_{-\infty}^\infty \vert x\vert f(x,ux)\,dx.$ (2)

For variates with a standard Normal Distribution, the ratio distribution is a Cauchy Distribution. For a Uniform Distribution
1 & for $x,y\in[0,1]$\cr
0 & otherwise,\cr}
\end{displaymath} (3)

0 & $u<0$\cr
\int_0^1 x\,dx=[{\textstyle{1\ov...
...xtstyle{1\over 2}}x^2]^{1/u}_0 ={1\over 2u^2} & for $u>1$.\cr}
\end{displaymath} (4)

See also Cauchy Distribution

© 1996-9 Eric W. Weisstein