This paper by Wassily Hoeffding, published in May 1962, derives upper bounds for the probability that the sum \( S \) of \( n \) independent random variables exceeds its mean \( ES \) by a positive number \( nt \). The random variables are assumed to have bounded or bounded above ranges. The bounds depend only on the endpoints of the ranges and the mean or the mean and the variance of \( S \). These results are then used to derive analogous inequalities for certain sums of dependent random variables, such as U statistics and the sum of a random sample without replacement from a finite population. The paper includes proofs and extensions of these inequalities, providing valuable tools for statistical analysis and probability theory.This paper by Wassily Hoeffding, published in May 1962, derives upper bounds for the probability that the sum \( S \) of \( n \) independent random variables exceeds its mean \( ES \) by a positive number \( nt \). The random variables are assumed to have bounded or bounded above ranges. The bounds depend only on the endpoints of the ranges and the mean or the mean and the variance of \( S \). These results are then used to derive analogous inequalities for certain sums of dependent random variables, such as U statistics and the sum of a random sample without replacement from a finite population. The paper includes proofs and extensions of these inequalities, providing valuable tools for statistical analysis and probability theory.