= Largest component of a Poisson-Dirichlet partition
{title2=$F_\theta(a)=\mathbb P(p_1\le a)$}
For $a\ge1/2$, at most one component exceeds $a$, so $1-F_\theta(a)=\theta\int_a^1x^{-1}(1-x)^{\theta-1}dx$. Below one half several components can exceed the threshold. Put $N_a=\#\{i:p_i>a\}$ and use the finite identity $\mathbf1_{\{N_a=0\}}=\sum_m(-1)^m(N_a)_m/m!$, with the <factorial moment densities of Poisson-Dirichlet components>. It gives the full <probability distribution> as finite <inclusion-exclusion> over $x_i>a$, $\sum_i x_i<1$. In contrast, a size-biased component is beta-distributed.
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