Let $X_1,\dots,X_n$ denote i.i.d. random variables that have bounded support. Denote by $X_{k:n}$ the $k$th order statistic, and $\mu_{k:n}^{r}=\mathbb{E}[X_{k:n}^r]$ the $r$th moment of this random variable.
What are some distributions with bounded support for which $\mu_{k:n}^{1}$ and $\mu_{k:n}^{2}$ admit a closed form characterization?
As an example consider the uniform distribution. Then, $X_{k:n}\sim \mathrm{Beta}(k,n+1-k)$. Thus, $\mu_{k:n}^{1} = \frac{k}{n+1}$ and $\mu_{k:n}^{2}= \frac{k(k+1)}{(n+1)(n+2)}$.
What are other bounded support distributions (or wrapped distributions) that admit similar closed-form characterizations?
