Uniform distribution on a simplex via i.i.d. random variables

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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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.