The goal is to end up with a ordered vector $v=(v_1,\dots,v_n)\in \mathbb{R}^n$ of $n$ random draws $v_1\leq v_2\dots\leq v_n$.
The random draws $v_i$ are iid, following a probability distribution with density $f$ mapping from the interval $I$ to $\mathbb{R}^+$ and with CDF $F$.
Does it matter whether I draw my vector directly from the set $\mathbb{O}=\{v\in\mathbb{R}|v_1\leq v_2\dots\leq v_n\}$ (with density functions potentially renormalized on the simplex) or whether I draw from the set $\mathbb{U}=I^n$ and sort the draws afterwards?
Would I end up with the same distribution of samples? Unfortunately, I do not have any idea how to adress this problem in general.
BONUS: Is the distribution of $v_1$ identical for both approaches? For the second approach, I know that the CDF of $v_1$ is $F_{v_1}(x)=1-[1-F(x)]^n$. Is this identical for the first case?