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If $S_n(p)$ has a binomial distribution with parameters $n,p$, there it is readily observed that ${\\rm argmax}_{0\\le p\\le 1}{\\mathbb E}S_n^2(p) = {\\rm argmax}_{0\\le p\\le 1}np(1-p) = \\frac12,$ and ${\\mathbb E}S_n^2(\\frac12) = \\frac{n}{4}$. Rabi Bhattacharya observed that while the second moment Chebyshev sample size for a $95\\%$ confi"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1711.07288","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2017-11-20T12:47:41Z","cross_cats_sorted":["math.ST","stat.TH"],"title_canon_sha256":"d50a9f07783618039dcfeb83b7e2862b93eaceba94f66c57f5d1c43aa58ece10","abstract_canon_sha256":"1ac21fe366ec3457ab93978d3727eb6408ad21089d9a6cab8bde8510b28620e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:30:10.594761Z","signature_b64":"imOcPEcHbiaV1QpwthUT7W78+AA7dzyT1J8EEmy12gI13h6GrDDhaZIxlTr5M+9fXqPJ2axPDmBcmA3ernFcBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f681a40dbafe1eb10816264174e911cad63969ae41b3d4d2b9dc6acb5a26fb20","last_reissued_at":"2026-05-18T00:30:10.594106Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:30:10.594106Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"When Fourth Moments Are Enough","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.ST","stat.TH"],"primary_cat":"math.PR","authors_text":"Chris Jennings-Shaffer, Dane R. Skinner, Edward C. Waymire","submitted_at":"2017-11-20T12:47:41Z","abstract_excerpt":"This note concerns a somewhat innocent question motivated by an observation concerning the use of Chebyshev bounds on sample estimates of $p$ in the binomial distribution with parameters $n,p$. Namely, what moment order produces the best Chebyshev estimate of $p$? If $S_n(p)$ has a binomial distribution with parameters $n,p$, there it is readily observed that ${\\rm argmax}_{0\\le p\\le 1}{\\mathbb E}S_n^2(p) = {\\rm argmax}_{0\\le p\\le 1}np(1-p) = \\frac12,$ and ${\\mathbb E}S_n^2(\\frac12) = \\frac{n}{4}$. 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