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.