the question is:
$$ X_1, X_2, ..., X_n $$ i.i.d random variables of uniform distribution U(a,b), where a<b. Show that $$ Z_i = \frac{X_{(i)}-X_{(1)}}{X_{(n)} - X_{(1)}}$$ ,i = 2,...n-1, are independent of $$(X_{(1)}, X_{(n)})$$, where $$X_{(i)}$$ are order statistics.
My basic idea is that: $$Z_i$$ is related to 3 order statistics: $$X_{(i)}, X_{(1)}, X_{(n)}$$. We may need to calculate the joint distribution of 3 order statistics, and then get the distribution of $$Z_i$$, and then check whether it's independent of $$X_{(1)}, X_{(n)}$$.
But it may be too complicate, are the any simpler solutions? Thank you very much.