= Mean measure of Poisson-Dirichlet components
{title2=$\alpha_\theta(dx)=\theta x^{-1}(1-x)^{\theta-1}\,dx$}
For $PD(\theta)$, $\mathbb E\sum_i\varphi(p_i)=\int\varphi\,d\alpha_\theta$ whenever the absolute integral is finite. Apply the <Mecke formula for a Poisson random measure> to a selected jump $u$ and the remaining total $t$, whose density is $t^{\theta-1}e^{-t}/\Gamma(\theta)$. Changing variables to $s=u+t$ and $x=u/s$ factors out a <Gamma function> and gives the displayed measure. Weighting it by $x$ gives a <probability> density $\theta(1-x)^{\theta-1}$, the <Beta distribution> of a size-biased component.
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