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

12/01/2017

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by   Benjamin Doerr, et al.

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We give an elementary proof of the fact that a binomial random variable X with parameters n and 0.29/n < p < 1 with probability at least 1/4 strictly exceeds its expectation. We also show that for 1/n < 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.