 One-parameter Poisson-Dirichlet distribution

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 $T={\sum }_i{J}_i$ 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.