17
$\begingroup$

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.)

$\endgroup$
2

1 Answer 1

16
$\begingroup$

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.

$\endgroup$
2
  • 1
    $\begingroup$ Very interesting. $\endgroup$
    – joriki
    Apr 12, 2011 at 21:43
  • $\begingroup$ Funny! I was already thinking about using the Gamma distribution because of its "infinite divisibility" $\endgroup$ Apr 12, 2011 at 22:32

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .