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Given some $s \in {[1,n]}$ and $n \in N$, $n\ge1$, what is the distribution of $X_i$ given that $\sum_{k=1}^{n} kX_k = s$, $\sum_{k=1}^{n} X_k = 1$, and $X_i \ge 0$.

If this is not a known distribution, how can I calculate the pdf/cdf?

Edit:

Thanks to whuber's help, I started looking into Dirichlet distribution. Indeed the $X_i$'s are intended to be an $(n-1)$-simplex with a flat Dirichlet distribution and a linear constraint. Here's what I did:

For simplicity, let's assume $n=3$. The linear constraint in this case is:

$X_1 + 2X_2 + 3X_3 = s \Rightarrow X_1 = s - 2X_2 - 3X_3$

Substituting $X_1$, the simplex constraint becomes:

$X_1+X_2+X_3 = s-X_2-2X_3 = 1$

or

$\frac{X_2}{s-1} + \frac{2X_3}{s-1} = 1$

A Dirichlet distribution can now used on the $1$-simplex $(u_1,u_2)$ by setting $u_1 = \frac{X_2}{s-1}$ and $u_2 = \frac{2X_3}{s-1}$. But unfortunately, the constraint $X_1\ge0$ is now lost (e.g., setting $(u_1,u_2)=(1,0)$ results in $X_1=s-2(s-1)=2-s$ which is negative when $2<s\le3$).

So I'm not sure if it is possible to linearly constrain a Dirichlet distribution and obtain another. Any help if you think it is possible (or confirm that it is not) would be appreciated.

omasoud
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    There isn't anywhere near enough information to answer this question, assuming the usual meanings of the notation: that is, that the $X_i$ are independent identically distributed random variables. What does the unusual expression "$R^{[1,n]}$" mean? What does it mean that "all combinations ... are equally likely"? What exactly is a "combination" in this context? – whuber Oct 14 '17 at 19:46
  • I edited the question and added some clarification. By combinations, I meant the set of $X_i$ values. For example, if $s=2.5, n=3$, $(X_1,X_2,X_3)=(0.25,0.00,0.75)$ is a combination that is equally likely to $(0.00,0.50,0.50)$ – omasoud Oct 14 '17 at 20:15
  • This sounds similar to https://stats.stackexchange.com/questions/214639. Closely related threads are https://stats.stackexchange.com/questions/33685 and perhaps within the links at https://stats.stackexchange.com/search?q=distribution+simplex. – whuber Oct 14 '17 at 22:19

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