On a Characterization of Uniform Distributions

Denoting by X(1) ≤ X(2) ≤ · · · ≤ X(n) the order statistic based on a random sample X1, X2, ..., Xn drawn from a distribution F, it is shown that the property "E(X1|X(1), X(n)) = 12(X(1) + X(n)), almost surely" holds for some n ≥ 3 if and only if F is either a uniform distribution on an interval or a discrete uniform distribution supported on a set of equispaced points. The heart of our proof is the interesting observation that the above equation involving conditional expectations completely determines the (topological) structure of the support of the underlying distribution F.