Random vector with a fixed sum: Distribution for its components

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.

• 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? – whuber Jun 10 at 20:47
• Thanks @wuber, I have realized if I do not add any constraints, except sum = 1, then they follow Dirichlet(1, 1, ..., 1). – TrungDung Jun 11 at 7:41
• @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. – TrungDung Jun 11 at 9:37
• Dirichlet$(\alpha,\alpha,\ldots,\alpha)$ distributions are all exchangeable. BTW, nothing I wrote implies the solution requires $\alpha=1.$ – whuber Jun 11 at 11:05
• I have a typo that makes a wrong statement. – TrungDung Jun 11 at 11:18