Expected value of $k$th ordered statistic in Uniform(0, r) for r<1 Suppose that we draw $X_1, \ldots, X_n$ independently and uniformly from $(0,r)$ and let $X_{(k)}$ denote the $k$th smallest number drawn.
Denoting the pdf of $X_{(k)}$ by function $f_k$, I know that
$$
f_{k}(x) = n \frac{1}{r} \binom{n-1}{k-1}(x/r)^{k-1}(1-x/r)^{n-k}.
$$
Questions


*

*How can I find the expected value of $X_{(k)}$?


I know that $\mathbb{E}[X_{(k)}] = \int_0^rf_k(x)xdx$; but I don't know how to solve it. Intuitively, I believe the answer should be $\mathbb{E}[X_{(k)}]=\frac{k}{n+1}r$ but I don't have any proofs for it.


*

*How concentrated is $X_{(k)}$ around its expectation? More precisely, I would like to know an upper bound on $\Pr[|X_{(k)}-\mathbb{E}[X_{(k)}]| > \epsilon]$ for given $\epsilon$.

 A: Here is an intuitive proof by contradiction of the first question without any Beta distribution. I will work with the case $r=1$ since for general $r$ you can just rescale.
So suppose without loss of generality that $\mathbb{E} X_{(k)} > \frac{k}{n+1}$. What this means is that if you draw $n$ points on $[0, 1]$ uniformly at random, repeatedly for $m$ times, eventually as $m \to \infty$ you will see points more concentrated on the right of $\mathbb{E} X_{(k)}$ than on its left. But this contradicts the assumption of uniform sampling on $[0, 1]$.
For the second question, you can get various bounds through the 2nd, 3rd, 4th, etc moments of Beta distributions, or their moment generating functions, as listed in the wikipedia page.
A: Drawing $X_1,\dots,X_n$ independently and uniformly from $(0,2\pi)$ can be done like this:
Choose $Z_1,\dots, Z_n, Z_{n+1}$ independently and uniformly on $S_1=\{(x,y)\mid x^2+y^2\}$. 
Then let $X_i$ be the length of the arc that connects $Z_{n+1}$ and $Z_i$ anti-clockwise.
The circle is split up in $n+1$ disjoint arcs and the setup reveals that the lengths of these arcs have equal distribution hence equal expectation.
That gives expectation $\frac{k}{n+1}\cdot2\pi$ for the length of the arc corresponding with $X_{(k)}$.
