An Elementary Analysis of the Probability That a Binomial Random Variable Exceeds Its Expectation
This addresses a theoretical probability problem for mathematicians and statisticians, offering incremental improvements in bounds for binomial distributions.
The paper tackles the problem of determining the probability that a binomial random variable exceeds its expectation, providing an elementary proof that for parameters n and p with 0.29/n ≤ p < 1, this probability is at least 1/4, and for 1/n ≤ p < 1 - 1/n, it exceeds by more than one with probability at least 0.0370, with both probabilities approaching 1/2 under certain conditions.
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.