14
$\begingroup$

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
15
$\begingroup$

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 '11 at 21:43
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.