The general case

Suppose now that $X_1,\dots,X_n$ is a random sample from a continuous distribution with pdf $f(x)$ and distribution function $F(x)$. Let $X_{(1)} \le,\dots,\le X_{(n)}$ be the ordered values.

Theorem 2   The joint pdf of $X_{(1)} \le,\dots,\le X_{(n)}$ is

$$f(x_{(1)},\dots,x_{(n)}) = n! \prod_{j=1}^n f\left(x_{(j)}\right) \;\;$$   for$$\;\; x_{(1)} \le,\dots,\le x_{(n)}.$$

The marginal pdf of $X_{(k)}$ is

$$f_{k}(x) = F(x)^{k-1}f(x) (1-F(x))^{n-k} \frac{n!}{(k-1)!(n-k)!},$$

and the joint pdf of $X=X_{(j)}$ and $Y=X_{(k)}$, where $j < k$ is

$$f_{jk}(x,y) = F(x)^{(j-1)}f(x) (F(y)-F(x))^{(k-j-1)}f(y) (1-F(y))^{(n-k)} \frac{n!}{(j-1)!(k-j-1)!(n-k)!},$$

on the set $x<y$.

Proof: This follows from Theorem 1. Just make the transformation $U_j = F(X_j)$ for $j=1,\dots,n$, and apply the change of variable.