On Equal Distributions

Abstract

It is shown that two distributions both of which have a finite expectation are equal if and only if for every $n \geqq 1$ there exists $1 \leqq k \leqq n$ such that the $k$th order statistics from samples of size $n$ of each distribution have equal expectations. Similarly, it is shown that a distribution with finite expectation is symmetric about zero if and only if for every $n \geqq 0$ there exists $0 \leqq k \leqq 2n + 1$ such that the sum of the expectations of the $k$th smallest and the $k$th largest observations in a sample of size $2n + 1$ is zero.