Suppose we have a parameter that follows some probability distribution $f(x)$. When simulating an $N$-body with that parameter as an attribute, how should values of the parameter be chosen?
Let each parameter be $x_1,...,x_N$, I figured that the expected value of each parameter could be calculated as follows. $$\mathbb{E}[x_i] = \int^1_0 d x_n\ ... \int^1_0 d x_1 x_i F(x_1,...x_n)$$ where $$F(x_1,...,x_n) = \left\{ \begin{array}{ll} n! f(x_1) ... f(x_n) & (0 \le x_1 \le ... \le x_n \le 1) \\ 0 & (\text{otherwise}) \end{array} \right.$$ $F$ is a conditional probability distribution because parameters are not distinguishable each other.
However, I cannot find any mathematical reason whether this is reasonable. If it is correct, please explain why. If not, please give me useful references.
