Handy Formulas for Binomial Moments
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Publication:6355815
arXiv2012.06270MaRDI QIDQ6355815
Maciej Skorski
Publication date: 11 December 2020
Abstract: Despite the relevance of the binomial distribution for probability theory and applied statistical inference, its higher-order moments are poorly understood. The existing formulas are either not general enough, or not structured and simplified enough for intended applications. This paper introduces novel formulas for binomial moments, in form of emph{polynomials in the variance} rather than in the success probability. The obtained formulas are arguably better structured, simpler and superior in their numerical properties compared to prior works. In addition, the paper presents algorithms to derive these formulas along with working implementation in the Python symbolic algebra package. The novel approach is a combinatorial argument coupled with clever algebraic simplifications which rely on symmetrization theory. As an interesting byproduct we establish emph{asymptotically sharp estimates for central binomial moments}, improving upon partial results from prior works.
Has companion code repository: https://github.com/maciejskorski/binomial_moments
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