= Beta-gamma independence
{title2=$X/(X+Y)\mathrel{\perp}(X+Y)$}
If $X$ and $Y$ are <independent random variables> with <Gamma distributions> of shapes $a,b>0$ and the same rate $\theta>0$, then $U=X/(X+Y)$ has the <Beta distribution> with parameters $(a,b)$, and $V=X+Y$ has the <Gamma distribution> of shape $a+b$ and rate $\theta$. Moreover $U$ and $V$ are independent. The inverse <change of variables> $(x,y)=(uv,(1-u)v)$ has absolute <Jacobian determinant> $v$, which makes the <joint probability density> factorize on the product domain $0<u<1$, $v>0$. The common rate is essential for this factorization.
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