On the Exponential Probability Bounds for the Bernoulli Random Variables

TL;DR

This paper derives optimized exponential probability bounds for Bernoulli variables' deviations, introducing new bounds with specific coefficients that improve classical inequalities and extend to continuous cases.

Contribution

It introduces a new exponential bound structure for Bernoulli variables, optimizing coefficients to improve deviation probability estimates and extending to continuous distributions.

Findings

Established bounds with coefficients {1, 2} for classical Bernoulli problem.

Derived new bounds with coefficients {1, 2/(1+epsilon^2)} for general Bernoulli cases.

Extended the exponential bounds framework to continuous distributions.

Abstract

We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha, beta}. In order to optimize the bound we are interested to minimize linear coefficient alpha and to maximize exponential coefficient beta. Generally, the value of linear coefficient alpha may not be smaller than one. The following 2 settings were proved: 1) {1, 2} in the case of classical discreet problem as it was formulated by Bernoulli in the 17th century, and 2) {1, 2/(1+epsilon^2)} in the general discreet case with arbitrary rational p and epsilon. The second setting represents a new structure of the exponential bound which may be extended to continuous case.