Let $P=(x_1,\ldots,x_n)$ be a population consisting of $n\ge 2$ real numbers whose sum is zero, and let $k <n$ be a positive integer. We sample $k$ elements from $P$ without replacement and denote by $X_P$ the sum of the elements in our sample. In this article, using ideas from the theory of majorization, we deduce non-asymptotic lower and upper bounds on the probability that $X_P$ exceeds its expected value.

翻译：设总体 $P=(x_1,\ldots,x_n)$ 由 $n\ge 2$ 个总和为零的实数构成，并令 $k <n$ 为正整数。我们从 $P$ 中无放回地抽取 $k$ 个元素，并用 $X_P$ 表示样本中元素的总和。本文利用优超理论的思想，推导出 $X_P$ 超过其期望值的概率的非渐近上下界。