Say we have $U_1 \dots U_n$ i.i.d. random variables uniform on $[0,1]$ and $Y_1 \dots Y_{n+1}$ i.i.d. random variables distributed as $Y_i \sim Exp(1)$. I know that the joint distribution of the order statistics $(U_{(1)}, \dots, U_{(n)})$ is equal in distribution to $(\frac{Y_1}{\sum_i^{n+1} Y_i}, \frac{Y_1+Y_2}{\sum_i^{n+1} Y_i}, \dots, \frac{Y_1+\dots+Y_n}{\sum_i^{n+1} Y_i})$. I can prove the result using a transformation of variables, Jacobians, etc., but this is rather tedious. Is there a more elegant way of deriving this statement? Maybe something with poisson processes?
Relations between Order Statistics of Uniform RVs and Exponential RVs
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probability
probability-theory
stochastic-processes
probability-distributions