n random variables' sum equals 1 , and they belong to the same distribution and are restricted in [0,1] , what distribution do they obey? $ X_1 + X_2 + .. + X_n = 1 $
$X_i$ is drawn from same distribution and they are 
restricted to [0,1] range , what distribution do they obey?
 A: Dirichlet Distribution (per Wikipedia) is a generalization of the Beta distribution (which satisfies your requirements, when $n=2$):
$$f(x_1,...,x_n, \alpha_1, ..., \alpha_n) = \frac{1}{B({\bf \alpha})}\prod_{i=1}^{n}x_i^{\alpha_i-1}$$

* $x_i \in (0,1)$

* $\sum_{i=1}^{n} x_i = 1$

* $\alpha_i > 0$

* $B(\bf{ \alpha} )$ is the Beta function, a normalizing constant in this case
A: This is similar to a multi-dimensional generalization of the Beta distribution. There is no one distribution that covers all such cases, but a common distribution in these scenarios is the Dirichlet distribution of order $n$. Given a set of $n$ positive parameters $\alpha_i$, the p.d.f. of this distribution is
$$
f(x_1,x_2,\ldots,x_n) \propto \prod x_i^{\alpha_i-1}
$$
A: The conditions you give don't put many constraints on the random variables, other than exchangeability.  For instance, if you let $Z_1, \ldots, Z_n$ be any sequence of iid positive random variables, then 
$$
  X_k = \frac{Z_k}{Z_1 + \cdots Z_n}
$$
satisfies your requirement.  If the $Z_k$ are Gamma($\alpha$,1) then you'll get the Dirichlet($\alpha,\ldots,\alpha$) mentioned in other answers.
So does $(Z_1, \ldots, Z_n)$ conditioned on $Z_1 + \cdots Z_n = 1$, as long as the conditioning makes sense (and now we can let the distribution for $Z$ put positive probability on zero, too).  
Interestingly, there starts to be much nicer answers if $n=\infty$: see this paper and associated work.
