For which $N \in \mathbb{N}$ is there a probability distribution such that $\frac{1}{\sum_i X_i} (X_1, \cdots, X_{N+1})$ is uniformly distributed over the $N$-simplex? (Where $X_1, \cdots, X_{N+1}$ are accordingly distributed iid random variables.)
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asked Apr 12 '11 at 21:06
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$\begingroup$ Related. $\endgroup$ – Wok Apr 30 '13 at 11:49
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$\begingroup$ This is used a lot in "compositional data analysis" so you can search for books about that. See also stats.stackexchange.com/questions/33685/… $\endgroup$ – kjetil b halvorsen Jan 2 '17 at 16:02
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Take a look at the Wikipedia article on the Dirichlet distribution. In particular the Dirichlet distribution with $\alpha_i = 1$ for all $i$ is the uniform distribution on the simplex. Furthermore, the Dirichlet distribution can be generated by taking $X_1, \ldots, X_n$ to be independent gamma random variables with the right choice of paramters, and then $Y_i = X_i/(X_1 + \cdots + X_n)$. In the particular case you're asking about, you can take the $X_i$ to all be exponential random variables with the same mean.
answered Apr 12 '11 at 21:28
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$\begingroup$ Funny! I was already thinking about using the Gamma distribution because of its "infinite divisibility" $\endgroup$ – charles.y.zheng Apr 12 '11 at 22:32
