Find the Expectation of uniform random variables

Question

I need to find the expectation of the following problem, but now assuming that U is uniformly distributed from a to b i.e U~(a,b). My problem is that I don't understand how to obtain the limits of integration.

Answer 1

Note that $U=(b-a)X+a$, where $X$ is uniform on $(0,1)$.

So the max of the $U_i$ is $a$ plus $(b-a)$ times the maximum of the corresponding $X_i$, and the same observation holds for the minimum. So the difference is $(b-a)$ times the corresponding difference in the $X_i$ world, and the expectation is $(b-a)\frac{n-1}{n+1}$.

Alternately we could compute the mean of the max of the $U_i$, and of the min, without going to the joint distribution. For in general $E(S-T)=E(S)-E(T)$.