Order Statistics of Independent Uniform Variates

Scenario In quantitative trading and quantitative research, candidates are frequently evaluated on probability modeling, specifically regarding Order Statistics / Beta Distribution.

Your Task
Let $U_1, U_2, \dots, U_n$ be independent and identically distributed $\text{Uniform}(0, 1)$ random variables. Let $U_{(1)} \le U_{(2)} \le \dots \le U_{(n)}$ denote their order statistics.
(1) Derive the probability density function $f_{U_{(k)}}(u)$ of the $k$-th order statistic.
(2) Compute $\mathbb{E}[U_{(k)}]$ and $\text{Var}(U_{(k)})$.

Constraints

• Samples are independent and identically distributed on their respective support.
• Boundary conditions are defined strictly within standard real intervals.