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Absolute moments of the binomial distribution folded at its mean
Abstract
Let X∼ Bin(N,p) and Y=|X−Np|. We derive an exact reduction for every odd absolute moment of Y, expressing it through finitely many central masses and central tail probabilities. The tail coefficients satisfy a sum rule and are divisible by q−p, so that in the symmetric case the odd ladder collapses to masses alone. We then obtain the complete central-tail expansion at a bounded lattice displacement, with Bernoulli-polynomial coefficients in integer powers of the large parameter. As an application, the first two tail coefficients yield a second-order expansion for the median of the beta distribution, whose formal one-parameter degeneration reproduces the first two terms of Choi's expansion for the gamma median.
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BibTeX
@misc{elezovic2026absolute,
title = {{Absolute moments of the binomial distribution folded at its mean}},
author = {Neven Elezović},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04416},
eprint = {2609.04416},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04416},
}
RIS
TY - UNPB
TI - Absolute moments of the binomial distribution folded at its mean
AU - Neven Elezović
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.04416
UR - https://arxiv.org/abs/2609.04416
AB - Let X∼ Bin(N,p) and Y=|X−Np|. We derive an exact reduction for every odd absolute moment of Y, expressing it through finitely many central masses and central tail probabilities. The tail coefficients satisfy a sum rule and are divisible by q−p, so that in the symmetric case the odd ladder collapses to masses alone. We then obtain the complete central-tail expansion at a bounded lattice displacement, with Bernoulli-polynomial coefficients in integer powers of the large parameter. As an application, the first two tail coefficients yield a second-order expansion for the median of the beta distribution, whose formal one-parameter degeneration reproduces the first two terms of Choi's expansion for the gamma median.
ER -