{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2020:ZM4NFLHW5C3C2J52EVMKU5GHQY","short_pith_number":"pith:ZM4NFLHW","schema_version":"1.0","canonical_sha256":"cb38d2acf6e8b62d27ba2558aa74c7861eaf445d375878f541fc05f7499b14f5","source":{"kind":"arxiv","id":"2006.00516","version":6},"attestation_state":"computed","paper":{"title":"Tight Probability Bounds with Pairwise Independence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.OC","authors_text":"Arjun Ramachandra, Karthik Natarajan","submitted_at":"2020-05-31T13:12:23Z","abstract_excerpt":"While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided in closed-form for any input marginal probability vector $\\mathbf{p} \\in [0,1]^n$. To prove the result, we show the existence of a positively correlated Bernoulli random vector with transformed bivariate pr"},"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":"2006.00516","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.OC","submitted_at":"2020-05-31T13:12:23Z","cross_cats_sorted":["math.CO","math.PR"],"title_canon_sha256":"a51052cb97d617cc03403bef6e0fa5bf06a8df3b454058cb0311c3da67b0aedc","abstract_canon_sha256":"f43c0fcfab01e3e2e2896f69e18a052d0fb4af7622cc56041c6dd060c4e1ede4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T05:18:42.633364Z","signature_b64":"azOCBcPvjVodz1S9pYZGdSppCE46m2TJ6GEIYRnvb8Io3hTASuTWt/NPapAwHFZHYOjJPIIvuFIwDrdETr99Dw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb38d2acf6e8b62d27ba2558aa74c7861eaf445d375878f541fc05f7499b14f5","last_reissued_at":"2026-07-05T05:18:42.632942Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T05:18:42.632942Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Tight Probability Bounds with Pairwise Independence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.PR"],"primary_cat":"math.OC","authors_text":"Arjun Ramachandra, Karthik Natarajan","submitted_at":"2020-05-31T13:12:23Z","abstract_excerpt":"While useful probability bounds for $n$ pairwise independent Bernoulli random variables adding up to at least an integer $k$ have been proposed in the literature, none of these bounds are tight in general. In this paper, we provide several results in this direction. Firstly, when $k = 1$, the tightest upper bound on the probability of the union of $n$ pairwise independent events is provided in closed-form for any input marginal probability vector $\\mathbf{p} \\in [0,1]^n$. To prove the result, we show the existence of a positively correlated Bernoulli random vector with transformed bivariate pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.00516","kind":"arxiv","version":6},"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/2006.00516/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2006.00516","created_at":"2026-07-05T05:18:42.632997+00:00"},{"alias_kind":"arxiv_version","alias_value":"2006.00516v6","created_at":"2026-07-05T05:18:42.632997+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.00516","created_at":"2026-07-05T05:18:42.632997+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZM4NFLHW5C3C","created_at":"2026-07-05T05:18:42.632997+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZM4NFLHW5C3C2J52","created_at":"2026-07-05T05:18:42.632997+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZM4NFLHW","created_at":"2026-07-05T05:18:42.632997+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY","json":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY.json","graph_json":"https://pith.science/api/pith-number/ZM4NFLHW5C3C2J52EVMKU5GHQY/graph.json","events_json":"https://pith.science/api/pith-number/ZM4NFLHW5C3C2J52EVMKU5GHQY/events.json","paper":"https://pith.science/paper/ZM4NFLHW"},"agent_actions":{"view_html":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY","download_json":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY.json","view_paper":"https://pith.science/paper/ZM4NFLHW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2006.00516&json=true","fetch_graph":"https://pith.science/api/pith-number/ZM4NFLHW5C3C2J52EVMKU5GHQY/graph.json","fetch_events":"https://pith.science/api/pith-number/ZM4NFLHW5C3C2J52EVMKU5GHQY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY/action/storage_attestation","attest_author":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY/action/author_attestation","sign_citation":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY/action/citation_signature","submit_replication":"https://pith.science/pith/ZM4NFLHW5C3C2J52EVMKU5GHQY/action/replication_record"}},"created_at":"2026-07-05T05:18:42.632997+00:00","updated_at":"2026-07-05T05:18:42.632997+00:00"}