Uniformized sample range ($F(X_{(n)})-F(X_{(1)})\sim \mathrm{Beta}(n-1,2)$)

For an iid sample of size $n\ge 2$ from a continuous cumulative distribution function, the probability integral transform makes the transformed sample uniform. Integrating the joint density of its minimum and maximum gives range density $n(n-1)r^{n-2}(1-r)$ on $0<r<1$. Its distribution function is $n{r}^{n-1}-(n-1)r^n$. The result describes population content between the extreme order statistics, independently of the underlying continuous distribution.