I am stuck with the following question:
There is a random variable $x\sim U[0,1] $. There are $n$ different draws (i.i.d.) being made. I am trying to compute the expected value/arithmetic mean of the $k$ highest of these draws.
That is, if I sort the draws $x_1\geq x_2\geq ...\geq x_n$, I am trying to compute $E[\frac{1}{k}\sum_{i=1}^kx_i]$.
While I have found this question on the expected value of the highest draw, I am unable to extend this to the average of the $k$ highes ones.
Add an $(n+1)$-th draw. Glue the ends of the interval together to form a circle, and cut the circle at the location of the $(n+1)$-th draw to form an interval again. The remaining $n$ draws are independently uniformly distributed on that interval. By symmetry, the $n+1$ segments between the $n+1$ draws all have the same expected length. Thus the expected value of $x_i$ is $1-\frac i{n+1}$, and the average you want is
$$ \frac1k\sum_{i=1}^k\left(1-\frac i{n+1}\right)=1-\frac{k+1}{2(n+1)}\;. $$
