Suppose we have random variables $Y_1,\dots,Y_n$ valued in $\mathbb{R}$. Denote the order statistics of $Y_1,\dots,Y_n$ as $Y_{(1)}\leq \dots \leq Y_{(n)}$. The order statistics has the distribution
$$f_{Y_{(1)},\dots, Y_{(n)}}(y_1,\dots, y_n)= n! \Pi_{i=1}^n f(y_i)$$
when $y_1\leq\dots \leq y_n$ and $0$ otherwise.
It is necessary that $Y_1,\dots,Y_n$ are independent and identical distributed? There might be a very easy way to see this but I don't have a clue.
No. Given independently identically distributed $Y_1, \ldots, Y_n$, define ${Y_1}'$ as $Y_1$ and ${Y_2}'$ to $Y'_n$ as the order statistics of $Y_2$ to $Y_n$. Then the order statistics of ${Y_1}'$ to ${Y_n}'$ have the given distribution, but ${Y_1}'$ to ${Y_n}'$ are not independently identically distributed.