= One-parameter Poisson-Dirichlet distribution
{title2=$PD(\theta)=PD(0,\theta)$}
Take a <Poisson random measure> on $(0,\infty)$ with intensity $\theta e^{-u}du/u$ and arrange its atoms as $J_1>J_2>\cdots$. The sum $T=\sum_iJ_i$ is finite and positive <almost surely>, has <Gamma distribution> with shape $\theta$ and rate one, and the law of $(J_i/T)$ is $PD(\theta)$. The order is decreasing, so the first component is the largest; a size-biased selection of a component has a different law.
Back to article page
