An Elementary Analysis of the Probability That a Binomial Random Variable Exceeds Its Expectation

TL;DR

This paper provides elementary proofs for the probability that a binomial random variable exceeds its expectation, establishing bounds and asymptotic behaviors for different parameter ranges.

Contribution

It offers new elementary proofs and bounds for the probability that a binomial variable exceeds its expectation, with explicit probability estimates.

Findings

Probability exceeds 1/4 for certain p ranges

Probability exceeds expectation by more than one with at least 0.0370

Probabilities approach 1/2 as np and n(1-p) grow large

Abstract

We give an elementary proof of the fact that a binomial random variable $X$ with parameters $n$ and $0.29/n \le p < 1$ with probability at least $1/4$ strictly exceeds its expectation. We also show that for $1/n \le p < 1 - 1/n$, $X$ exceeds its expectation by more than one with probability at least $0.0370$. Both probabilities approach $1/2$ when $np$ and $n(1-p)$ tend to infinity.