Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. The goal is to prove that:
$$ W = U_{(s)}-U_{(r)} \sim \textrm{Beta}(s-r, \, n - s + r +1) \qquad 1 \leq r < s \leq n $$
How do I prove this? I have tried the method of using the joint pdf and condensing it to the pdf of the difference to no avail. Thanks.