Distributions of the mean values for some random measures

Let $\tau$ be a probability measure on $[0,1]$. We consider a generalization of the classic Dirichlet process – the random probability measure $F=\sum P_i\delta_{X_i}$, where $X=\{X_i\}$ is a sequence of independent random variables with the common distribution $\tau$ and $P=\{P_i\}$ is independent of $X$ and has the two-parameter Poisson–Dirichlet distribution $PD(\alpha,\theta)$ on the unit simplex. The main result is the formula connecting the distribution $\mu$ of the random mean value $\int x\,dF(x)$ with the parameter measure $\tau$.