{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:MBXSYZF4Z7GKYGBEPS4PWUICVK","short_pith_number":"pith:MBXSYZF4","canonical_record":{"source":{"id":"1807.03981","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-11T07:59:24Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"0bf3dc63f4178ceab0bcf3c134cc0e1587e346c270c80f3107246b119ba391f6","abstract_canon_sha256":"4994c50d2e9ab01ede2f34baf4a0ebb479626eeed09c10bd693c00de52a2967a"},"schema_version":"1.0"},"canonical_sha256":"606f2c64bccfccac18247cb8fb5102aa898f47f3f99e558f228cd16ed8d46356","source":{"kind":"arxiv","id":"1807.03981","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.03981","created_at":"2026-05-17T23:49:27Z"},{"alias_kind":"arxiv_version","alias_value":"1807.03981v1","created_at":"2026-05-17T23:49:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.03981","created_at":"2026-05-17T23:49:27Z"},{"alias_kind":"pith_short_12","alias_value":"MBXSYZF4Z7GK","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MBXSYZF4Z7GKYGBE","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MBXSYZF4","created_at":"2026-05-18T12:32:37Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:MBXSYZF4Z7GKYGBEPS4PWUICVK","target":"record","payload":{"canonical_record":{"source":{"id":"1807.03981","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-11T07:59:24Z","cross_cats_sorted":["stat.TH"],"title_canon_sha256":"0bf3dc63f4178ceab0bcf3c134cc0e1587e346c270c80f3107246b119ba391f6","abstract_canon_sha256":"4994c50d2e9ab01ede2f34baf4a0ebb479626eeed09c10bd693c00de52a2967a"},"schema_version":"1.0"},"canonical_sha256":"606f2c64bccfccac18247cb8fb5102aa898f47f3f99e558f228cd16ed8d46356","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:49:27.640832Z","signature_b64":"7eNRZbq4PsLjBvifvIoyBzVw3ihyts4a8p3hpnnNFAJaqrgCkXE2RH3AZI8CAiws6fpDuJYTaNvelOFNSLTlAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"606f2c64bccfccac18247cb8fb5102aa898f47f3f99e558f228cd16ed8d46356","last_reissued_at":"2026-05-17T23:49:27.640246Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:49:27.640246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.03981","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:49:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZOJXuDyDJqTdMpxdfrV3oeYN3zDqY5QMLjSJK4oKRG770Gzi0NRqkjYqwTtFiwrxAQ1wxxPwjc0V9cYKZTWvDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:09:31.786854Z"},"content_sha256":"7154b43ec1fca17d6852b86128272b78a2b1130a0ddf22d2cd7a91a29009b2af","schema_version":"1.0","event_id":"sha256:7154b43ec1fca17d6852b86128272b78a2b1130a0ddf22d2cd7a91a29009b2af"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:MBXSYZF4Z7GKYGBEPS4PWUICVK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A note on the distribution of the product of zero mean correlated normal random variables","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["stat.TH"],"primary_cat":"math.ST","authors_text":"Robert E. Gaunt","submitted_at":"2018-07-11T07:59:24Z","abstract_excerpt":"The problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables dates back to 1936. Perhaps surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero mean correlated normal random variables as a variance-gamma random variable, from which an explicit formula for probability density function is immediate."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03981","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":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:49:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"UVJqG4VWxrYlaQXwX1syqU6ZBAtQ2m+wmgeWyAn1HwnhvdL3jG0U5Rd6IW0ZoErsy3ROMHohtBZS47YOEyr0CA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T20:09:31.787564Z"},"content_sha256":"a07b867e5ad24f5e6b608ba66ee8c2aca53a785d9266d5a68d6206b74b7da61e","schema_version":"1.0","event_id":"sha256:a07b867e5ad24f5e6b608ba66ee8c2aca53a785d9266d5a68d6206b74b7da61e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK/bundle.json","state_url":"https://pith.science/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T20:09:31Z","links":{"resolver":"https://pith.science/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK","bundle":"https://pith.science/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK/bundle.json","state":"https://pith.science/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MBXSYZF4Z7GKYGBEPS4PWUICVK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MBXSYZF4Z7GKYGBEPS4PWUICVK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"4994c50d2e9ab01ede2f34baf4a0ebb479626eeed09c10bd693c00de52a2967a","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-11T07:59:24Z","title_canon_sha256":"0bf3dc63f4178ceab0bcf3c134cc0e1587e346c270c80f3107246b119ba391f6"},"schema_version":"1.0","source":{"id":"1807.03981","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.03981","created_at":"2026-05-17T23:49:27Z"},{"alias_kind":"arxiv_version","alias_value":"1807.03981v1","created_at":"2026-05-17T23:49:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.03981","created_at":"2026-05-17T23:49:27Z"},{"alias_kind":"pith_short_12","alias_value":"MBXSYZF4Z7GK","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MBXSYZF4Z7GKYGBE","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MBXSYZF4","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:a07b867e5ad24f5e6b608ba66ee8c2aca53a785d9266d5a68d6206b74b7da61e","target":"graph","created_at":"2026-05-17T23:49:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"The problem of finding an explicit formula for the probability density function of two zero mean correlated normal random variables dates back to 1936. Perhaps surprisingly, this problem was not resolved until 2016. This is all the more surprising given that a very simple proof is available, which is the subject of this note; we identify the product of two zero mean correlated normal random variables as a variance-gamma random variable, from which an explicit formula for probability density function is immediate.","authors_text":"Robert E. Gaunt","cross_cats":["stat.TH"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-11T07:59:24Z","title":"A note on the distribution of the product of zero mean correlated normal random variables"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.03981","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7154b43ec1fca17d6852b86128272b78a2b1130a0ddf22d2cd7a91a29009b2af","target":"record","created_at":"2026-05-17T23:49:27Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"4994c50d2e9ab01ede2f34baf4a0ebb479626eeed09c10bd693c00de52a2967a","cross_cats_sorted":["stat.TH"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.ST","submitted_at":"2018-07-11T07:59:24Z","title_canon_sha256":"0bf3dc63f4178ceab0bcf3c134cc0e1587e346c270c80f3107246b119ba391f6"},"schema_version":"1.0","source":{"id":"1807.03981","kind":"arxiv","version":1}},"canonical_sha256":"606f2c64bccfccac18247cb8fb5102aa898f47f3f99e558f228cd16ed8d46356","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"606f2c64bccfccac18247cb8fb5102aa898f47f3f99e558f228cd16ed8d46356","first_computed_at":"2026-05-17T23:49:27.640246Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:49:27.640246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"7eNRZbq4PsLjBvifvIoyBzVw3ihyts4a8p3hpnnNFAJaqrgCkXE2RH3AZI8CAiws6fpDuJYTaNvelOFNSLTlAA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:49:27.640832Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.03981","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7154b43ec1fca17d6852b86128272b78a2b1130a0ddf22d2cd7a91a29009b2af","sha256:a07b867e5ad24f5e6b608ba66ee8c2aca53a785d9266d5a68d6206b74b7da61e"],"state_sha256":"8d23914617f515233312511d5a8755e412c9c9138a7225d7bf2acd75bb8dc214"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YUCHwsAs0ZH3dmJQ+cs+2b/bu+qWjCY0tz0+7zQE3ifqgiu3BhoVjGpEqgBQQbpewQo1R5LcpBfcOMLNcq1FAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T20:09:31.791976Z","bundle_sha256":"9d890cf4ff8d58beae8a443ff6608263d7afd6106d1ef8e58ae74a402eab312b"}}