Suppose that $(f_0,\dotsc,f_N)$, with $f_n\ge0, \sum_n f_n=1$, is a distribution (set of normalized weights or frequencies) having a Dirichlet distribution with parameters $\alpha_n$: $$\mathrm{p}(f_0,\dotsc,f_N) = \mathrm{Dir}(f_0,\dotsc,f_N \mid \alpha_0,\cdots, \alpha_N)$$ and suppose that the values $n\in\{0,\cdots,N\}$ are associated with the distribution $(f_n)$.
What is the distribution of the mean $m := \sum_{n=0}^N n\ f_n$?
(The expectation and variance of $m$ can be easily found via the formulae for the mean and variance of linear combinations of correlated random variables. But I'm interested in the full distribution.)
Explicit integration for the simple case $N=3$ seems to lead to a piecewise function involving hypergeometric functions. So maybe there's no general and simple analytic formula for the general $N$ case. But I prefer asking this community.
