= Beta sampling by gamma ratios
{title2=$S/(S+T)$}
Independent <gamma distributions> $S\sim\Gamma(\alpha,r)$ and $T\sim\Gamma(\beta,r)$ with a common rate yield a <Beta distribution> $S/(S+T)\sim\operatorname{Beta}(\alpha,\beta)$. The <change of variables> $S=wu$, $T=w(1-u)$ has <Jacobian determinant> $w$, and factors the joint <probability density function> into a <Beta distribution> density in $u$ and a <gamma distribution> density in $w$. In particular, independent <chi-squared distributions> with $2\alpha$ and $2\beta$ degrees of freedom give the required ratio.
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