= Beta-prime distribution
{title2=$f(x)=\frac{x^{a-1}}{B(a,b)(1+x)^{a+b}},\quad x>0$}
For $a,b>0$, this <probability density function> is obtained from a <beta distribution> variable $U$ by $X=U/(1-U)$. Indeed $u=x/(1+x)$ and $du/dx=(1+x)^{-2}$, and the <change of variables formula> gives the displayed density. In particular, the ratio of two <independent> equal-rate <exponential distributions> has $a=b=1$, density $(1+x)^{-2}$ and <cumulative distribution function> $x/(1+x)$.
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