Let $X_1, \ldots, X_n$ be $n$ i.i.d. variables with probability density $f(x)$. Let $X_{(1)}, \ldots, X_{(n)}$ be the ordered statistics. Then the joint probability density of all $n$ order statistics is:
$$f_{X_{(1)}, \ldots, X_{(n)}}(x_1, \ldots, x_n) = n! \prod_{i=1}^n f(x_i),~ x_1 \leq \ldots, x_n \tag{1}$$
Assume now that $Y_1, \ldots, Y_n$ are not identically distributed but still independent variables with probability densities respectively $f_i(x)$. What is the joint probability density $f_{Y_{(1)}, \ldots, Y_{(n)}}(y_1, \ldots, y_n)$ of all the order statistics for $Y_1, \ldots, Y_n$?
I wonder if eq. (1) can be generalized in intuitive way:
$$f_{Y_{(1)}, \ldots, Y_{(n)}}(y_1, \ldots, y_n) = n! \prod_{i=1}^n f_i(y_i),~ y_1 \leq \ldots y_n \tag{2}$$