{"paper":{"title":"Binomial probabilities at a fixed distance from the mode: size-biasing and the complete asymptotic expansion","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Neven Elezovi\\'c","submitted_at":"2026-07-22T07:30:15Z","abstract_excerpt":"Let $X\\sim Bin(N,p)$ with $0<p<1$ and $q=1-p$, let $\\nu=\\lceil Np\\rceil$ be the first lattice point not below the mean, and let $r$ be a fixed integer. We determine the complete asymptotic expansion, in powers of $N^{-1}$, of the binomial mass $\\Pr\\{X=\\nu+r\\}$ -- a quotient of gamma functions -- uniformly for $p$ in compact subintervals of $(0,1)$ and for bounded $r$. Because the mean $Np$ is not a lattice point, the coefficients cannot be constants: they are Bernoulli polynomials evaluated at the oscillating fractional displacement $h_N=\\nu-Np\\in[0,1)$, and are given in closed form to all ord"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.19844","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.19844/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}