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

$$D(u) = P(U\leq u)$$

$$= P(Y\leq uX \vert X>0)+P(Y\geq uX\vert X<0)$$

$$= \int_0^\infty \int_0^{ux} f(x,y)\,dy\,dx+\int_{-\infty}^0\int_{ux}^0 f(x,y)\,dy\,dx.$$ (1)

The probability function is then

$$P(u) = D'(u)=\int_0^\infty xf(x,ux)\,dx-\int_{-\infty}^0 xf(x,ux)\,dx$$

$$= \int_{-\infty}^\infty \vert x\vert f(x,ux)\,dx.$$ (2)

For variates with a standard Normal Distribution, the ratio distribution is a Cauchy Distribution. For a Uniform Distribution

$$f(x,y)=\cases{ 1 & for $x,y\in[0,1]$\cr 0 & otherwise,\cr}$$ (3)

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

See also Cauchy Distribution