Difference of order statistics in a sample of uniform random variables

Solution 1:

One does not need to evaluate the integral. The change of variables $u=(1-w)v$ yields that $f_W(w)$ is a constant factor $C$ times $w^{s-1-r}$ times $(1-w)^{n+r-s}$. This also yields the value of the constant factor $C$ since $f_W$ must integrate to $1$.

(Or, after the change of variables, one can recognize the integral over $v$ from $0$ to $1$ as a Beta.)

Solution 2:

Consider this alternative proof: One can generate $n$ uniform random variables on a line by the following equivalent way: 1. Generating $n+1$ uniform random variables on a circle. 2. Choosing a random point as a references point ('$0$' point).

Therefore $U(s)−U(r)$ has an identical distribution to $U(s-1)−U(r-1)$, and to $U(s-r)−U(0)=U(s-r)-0 = U(s-r)$. We already know $U(s-r) \sim \text{Beta}(s−r,n−(s+r)+1)$