Let $X_1, \cdots , X_n$ be identically distributed random variables, with probability function $~ Exp (θ)$ with $θ = 1$.
Let $Y_1, \cdots, Y_n$ be the corresponding order statistics:
Obtain the joint density function of $Y_1$, and of $Y_n$, then show that $nY_n$ and $n(Y_2-Y_1)$ are independent
What I have done:
Remembering that for a random sample whose order statistics are $X_{(1)} \cdots X_{(n)}$
Then the joint pdf of $X_{(i)}$ and $X_{(j)}$ is:
$f_{X(i), X(j)} (u,v) = \dfrac{n!}{(i-1)! (j-1-i)! (n-j)!} f_{X}(u) f_{X}(v) [F_{X}(u)]^{i-1} \times [F_{X}(v) - F_{X}(u)]^{j-1-i} [1 - F_{X}(v)]^{n-j} $
Then
% \begin{align}
$f_{Y(i), Y(j)} (u,v) = \dfrac{n!}{(1-1)! (n-1-1)! (n-n)!} f_{X}(u) f_{X}(v) [F_{X}(u)]^{1-1} \times [F_{X}(v) - F_{X}(u)]^{n-1-1} [1 - F_{X}(v)]^{n-n} $
$ = \dfrac{n!}{(n-2)!} f_{X}(u) f_{X}(v) \times [F_{X}(v) - F_{X}(u)]^{n-2} $
% \end{align}