Largest component of a Poisson-Dirichlet partition ($F_{\theta }(a)=\mathbb{P}(p_1\le a)$)

For $a\ge 1/2$, at most one component exceeds $a$, so $1-F_{\theta }(a)=\theta {\int }_a^1{x}^{-1}(1-x{)}^{\theta -1}dx$. Below one half several components can exceed the threshold. Put $N_a=\#\{i:p_i>a\}$ and use the finite identity $1_{\{N_a=0\}}={\sum }_m(-1{)}^m(N_a{)}_m/m!$, 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.