Let $ X = X_1, ..., X_n $ i.i.d random variables with $U_{[0, 1]}$ distribution. I need to prove that $X_{(k)} \sim \xi = \frac{\xi_1 + ... + \xi_{k - 1}}{\xi_1 + ... + \xi_{n+1}}$, where $\xi_i$ are independent with $Exp(1)$ distribution.
It's well known that $U_{(k)} \sim Beta(k, n + 1 - k)$ and $\sum_{i=1}^{n} \xi_i \sim \Gamma(n, 1)$ and its also easy to find distribution of ratio of independent Gamma distributions, however in our case Gamma distributions are dependent, and my question is how can we find distribution of $\xi$?