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Suppose that I have a random vector $X=(p_1, p_2, ..., p_n)$ where each $0\leq p_i\leq 1$ and $\sum p_i = 1$. I would like to find distribution for each component $p_i$ and also other general expressions such as $p_1+p_2+p_3+p_4$.

I have used function RandVec in R to generated one million vectors. Histograms for each $p_i$ can be seen here (for case $n=8$).

EDIT: Thank to @wuber, I have realized if I do not add any constraints, except sum = 1, then they follow Dirichlet(1, 1, ..., 1). Please correct me if this is incorrect.

enter image description here

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    $\begingroup$ A nice family of such distributions is the Dirichlet. You can also take (literally) any positive random variable $(q_1,\ldots, q_n)$ and normalize its values by their sum. But could you explain what you mean by "find" a distribution? Based on what criteria or information? $\endgroup$ – whuber Jun 10 at 20:47
  • $\begingroup$ Thanks @wuber, I have realized if I do not add any constraints, except sum = 1, then they follow Dirichlet(1, 1, ..., 1). $\endgroup$ – TrungDung Jun 11 at 7:41
  • $\begingroup$ @wuber: It seems that permutation of the component does not have the same distribution, even with Dir(1, ..., 1). Do you think so, although intuitively, I think it should be. $\endgroup$ – TrungDung Jun 11 at 9:37
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    $\begingroup$ Dirichlet$(\alpha,\alpha,\ldots,\alpha)$ distributions are all exchangeable. BTW, nothing I wrote implies the solution requires $\alpha=1.$ $\endgroup$ – whuber Jun 11 at 11:05
  • $\begingroup$ I have a typo that makes a wrong statement. $\endgroup$ – TrungDung Jun 11 at 11:18

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