This is an open question: given a sequence of $n$ real numbers $x_1<x_2<\dots<x_n$, does there always exist a probability distribution, such that $\{x_i\}$ happens to be the $n$ expected order statistics of this distribution?
In other words, can we always "reverse engineer" the distribution from its expected order statistics? Note that there is no restrictions on the distribution, i.e., it can be continuous, discrete, or whatever. I wonder if anything is known regarding this existence problem.
(Edited 09/20/2017)
When sequence $\{x_i\}$ is unrestricted, the answer to above claim is no. This is shown by @Mateusz Kwaśnicki when $n=4$. However, under $n=4$, suppose $\{x_i\}$ satisfies the condition that $3(x_4−x_1)⩽7(x_3−x_2)$, then is there a method that can construct the distribution for which $\{x_1,x_2,x_3,x_4\}$ are the expected order statistics?
In other words, suppose $\{x_i\}$ satisfies the necessary conditions to be expected order statistics, is there a method to "reconstruct" the underlying distribution? Or is this too much to ask for?