{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261028-r06-random-variable-independence-0072", "pilot_number": "R0027-14022", "number": 14022, "family_id": "qf-mod28-rvind-event-preimage", "title": { "zh-Hant": "第 14022 題：隨機變數獨立", "en": "Question 14022: A Selected Pair of Value Events under the Same Joint Law" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261028-r06-random-variable-independence-0072", "batch_id": "batch-20261014-advance-20261028-r06-random-variable-independence-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-random-variable-independence" ], "title_zh": "隨機變數獨立", "stem_zh": "全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[9,34],\"y_values\":[-3,41],\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"y_event\":[-3]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。", "answer": { "kind": "expression", "value": "([34], [-3], 2/7, 2/7, 4/49, 4/49, 是)", "explanation_zh": "Wave4第二層核對=([34], [-3], 2/7, 2/7, 4/49, 4/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。" }, "difficulty": { "band": "undergraduate_core", "level": 3, "skills": [ "隨機變數獨立" ] }, "variant": { "family_id": "qf-mod28-rvind-event-preimage", "seed": 202610286072, "parameters": { "x_values": [ 9, 34 ], "y_values": [ -3, 41 ], "masses": [ [ [ 10, 49 ], [ 25, 49 ] ], [ [ 4, 49 ], [ 10, 49 ] ] ], "x_event": [ 34 ], "y_event": [ -3 ], "domain": "Exact_finite_joint_random_variable_independence", "operation": "rvind-event-preimage", "semantic_template": "qf-mod28-rvind-event-preimage/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"x_values\":[9,34],\"y_event\":[-3],\"y_values\":[-3,41]}" } }, "verification": { "status": "candidate", "checks": [ "producer_exact_finite_model", "producer_source_snapshot_check", "producer_substantive_template_deduplication", "producer_json_schema" ], "warning_zh": "產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。" }, "provenance": { "generator_role": "question_producer", "generator_model": "model_unknown", "generated_at": "2026-10-14", "contract_version": "ocme-question-factory-v0.1" } }, "original_english_edition": { "schema_version": "ocme-question-localization-v0.1", "question_id": "q-20261014-advance-20261028-r06-random-variable-independence-0072", "batch_id": "batch-20261014-advance-20261028-r06-random-variable-independence-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "52144ae39f0abe7a2ec02ea1690c5c8584445ea9dc3ca4154bec7be80b9a3ea0", "family_id": "qf-mod28-rvind-event-preimage", "seed": 202610286072, "semantic_template": "qf-mod28-rvind-event-preimage/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"x_values\":[9,34],\"y_event\":[-3],\"y_values\":[-3,41]}" }, "fields": { "title": "A Selected Pair of Value Events under the Same Joint Law", "stem": "Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[34], y_event=[-3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?).", "answer": { "kind": "expression", "value": "([34], [-3], 2/7, 2/7, 4/49, 4/49, Yes)", "explanation": "Aggregate marginal event masses and the corresponding joint rectangle exactly. This checks the two selected inverse-image events, not all event choices by itself. An exact finite or named-formula slice does not automatically establish an arbitrary infinite theorem. The exact result, retaining the requested order, is ([34], [-3], 2/7, 2/7, 4/49, 4/49, Yes)." }, "skills": [ "A Selected Pair of Value Events under the Same Joint Law" ], "verification_warning": "This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review." } }, "givens": [ { "id": "G001", "text": { "zh-Hant": "全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[9,34],\"y_values\":[-3,41],\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"y_event\":[-3]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。", "en": "Use the complete joint grid with original x_values=[9, 34], y_values=[-3, 41], and aligned row-by-column exact masses=[[10/49, 25/49], [4/49, 10/49]]. Value labels must be distinct in each nonempty list, all masses nonnegative and their total exactly 1. Preserve grid order and zero masses; all marginals, events and moments use this same law. Two-variable independence requires factorization for every grid pair; it implies neither identical distribution, IID, Markov behavior nor larger-family mutual independence. Use exact reduced rational outputs with positive denominators; omit /1 for integers. x_event=[34], y_event=[-3] must use labels from their respective variable ranges. Normalize them as increasing distinct sets. Give (x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal?)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "joint grid model whole mass sum := sum(10/49, 25/49, 4/49, 10/49)", "after": "joint grid model whole mass sum = 1", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 joint grid model whole mass sum，不以小數近似替代。", "en": "Recompute joint grid model whole mass sum with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "10/49", "25/49", "4/49", "10/49" ], "output": "1", "label": "joint grid model whole mass sum" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "joint grid model:raw aligned masses=[[10, 49], [25, 49], [4, 49], [10, 49]]", "after": "normalized masses=[10/49, 25/49, 4/49, 10/49];nonnegative=是;total=1", "relation": "implies", "rule_id": "whole_finite_probability_law_not_uniform_assumption", "conditions": [], "explanation": { "zh-Hant": "完整同一有限機率模型先核非負與總和恰為1；零質量原子仍保留，不默認均勻或另換模型。", "en": "First qualify nonnegativity and exact unit total on one complete finite probability model.Retain zero-mass atoms;do not assume uniformity or substitute another model." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S003", "uses": [ "G001", "S002" ], "before": "original x row labels=[9, 34];original y column labels=[-3, 41]", "after": "normalized aligned row-by-column joint grid=[[10/49, 25/49], [4/49, 10/49]]", "relation": "implies", "rule_id": "RV_joint_grid_original_order_not_value_sort", "conditions": [], "explanation": { "zh-Hant": "值清單各自不重複，但可非數值排序。所有質量依原行列位置配對，包含零格，不按值排序後錯配矩陣。", "en": "Each value list is distinct but need not be numerically sorted.Pair every mass with its original row/column,including zero cells;sorting values must not misalign the grid." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S004", "uses": [ "G001", "S003" ], "before": "X row0 marginal := sum(10/49, 25/49)", "after": "X row0 marginal = 5/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 X row0 marginal，不以小數近似替代。", "en": "Recompute X row0 marginal with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "10/49", "25/49" ], "output": "5/7", "label": "X row0 marginal" } }, { "id": "S005", "uses": [ "G001", "S004" ], "before": "X row1 marginal := sum(4/49, 10/49)", "after": "X row1 marginal = 2/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 X row1 marginal，不以小數近似替代。", "en": "Recompute X row1 marginal with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4/49", "10/49" ], "output": "2/7", "label": "X row1 marginal" } }, { "id": "S006", "uses": [ "G001", "S005" ], "before": "Y column0 marginal := sum(10/49, 4/49)", "after": "Y column0 marginal = 2/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Y column0 marginal，不以小數近似替代。", "en": "Recompute Y column0 marginal with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "10/49", "4/49" ], "output": "2/7", "label": "Y column0 marginal" } }, { "id": "S007", "uses": [ "G001", "S006" ], "before": "Y column1 marginal := sum(25/49, 10/49)", "after": "Y column1 marginal = 5/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 Y column1 marginal，不以小數近似替代。", "en": "Recompute Y column1 marginal with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "25/49", "10/49" ], "output": "5/7", "label": "Y column1 marginal" } }, { "id": "S008", "uses": [ "G001", "S007" ], "before": "raw x event=[34];raw y event=[-3]", "after": "normalized events=[[34], [-3]];original matching rows=[1];columns=[0]", "relation": "implies", "rule_id": "RV_value_event_range_guard_and_grid_indices", "conditions": [], "explanation": { "zh-Hant": "值事件去重排序，但必須屬各自宣告值域；再回原grid索引取質量，不能把值當列號或從別個模型拿邊際。", "en": "Normalize value events as sets and require their declared ranges.Then use original grid indices,not value-as-index lookups or marginals from another model." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null }, { "id": "S009", "uses": [ "G001", "S008" ], "before": "selected X event mass := sum(2/7)", "after": "selected X event mass = 2/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected X event mass，不以小數近似替代。", "en": "Recompute selected X event mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/7" ], "output": "2/7", "label": "selected X event mass" } }, { "id": "S010", "uses": [ "G001", "S009" ], "before": "selected Y event mass := sum(2/7)", "after": "selected Y event mass = 2/7", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected Y event mass，不以小數近似替代。", "en": "Recompute selected Y event mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "2/7" ], "output": "2/7", "label": "selected Y event mass" } }, { "id": "S011", "uses": [ "G001", "S010" ], "before": "selected joint event rectangle mass := sum(4/49)", "after": "selected joint event rectangle mass = 4/49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected joint event rectangle mass，不以小數近似替代。", "en": "Recompute selected joint event rectangle mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sum", "inputs": [ "4/49" ], "output": "4/49", "label": "selected joint event rectangle mass" } }, { "id": "S012", "uses": [ "G001", "S011" ], "before": "selected event marginal product := mul(2/7, 2/7)", "after": "selected event marginal product = 4/49", "relation": "calculation", "rule_id": "exact_rational_arithmetic", "conditions": [], "explanation": { "zh-Hant": "依原始角色、完整配對與已核中間量精確計算；保留分母、正負號及零值。", "en": "Compute exactly from original roles,complete pairings and established intermediates,retaining denominators,signs and zeros." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "以精確有理數重算 selected event marginal product，不以小數近似替代。", "en": "Recompute selected event marginal product with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "mul", "inputs": [ "2/7", "2/7" ], "output": "4/49", "label": "selected event marginal product" } }, { "id": "S013", "uses": [ "G001", "S012" ], "before": "P_X_event=2/7;P_Y_event=2/7;P_joint=4/49;product=4/49", "after": "selected event pair factorizes=是;not an all-event independence certificate", "relation": "implies", "rule_id": "RV_one_event_pair_not_all_events", "conditions": [], "explanation": { "zh-Hant": "單一事件對通過不取代所有值對分解；例如全集或空事件可能在相依模型也通過。", "en": "One passing event pair does not replace all-cell factorization.Full or empty events may pass even in a dependent model." }, "reverse_audit": { "kind": "premise_and_result_audit", "explanation": { "zh-Hant": "回到原始資料與本步條件核對，不把單向推論誤稱等價。", "en": "Check against the original data and this step's premises; a one-way implication is not an equivalence." } }, "arithmetic": null } ], "math_model": { "result": [ [ 34 ], [ -3 ], "2/7", "2/7", "4/49", "4/49", true ] }, "conclusion": { "canonical_value": "([34], [-3], 2/7, 2/7, 4/49, 4/49, 是)", "explanation": { "zh-Hant": "依上列完整代值及條件核對得到原題答案；原題編號、格式與狀態保持不變。", "en": "The full substitutions and premise checks give the original answer; source identity,format and status are unchanged." } }, "definition_note": { "zh-Hant": "沿用原題明示的論域、假設與來源 MKO 定義。本題已完成第二對話數理與中英語義覆核，並由原始參數逐題獨立重算模型、答案與每個算術步驟；原題仍為 candidate。", "en": "Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate." }, "source": { "input_commit": "0f7be61e2c22317a4075561dc822c87d881a50db", "question_binding": { "batch_id": "batch-20261014-advance-20261028-r06-random-variable-independence-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261028-r06-random-variable-independence-foundations/questions.jsonl", "batch_sha256": "d01de33201486c30f18c651effaac5b2d50d64207ccccd8f6c9125ce3a4a6c02", "original_line": 72, "original_line_sha256": "613546339176a9daa5801a730a9f425ad9c30125b15e0b7823e5a6d94c1b7a24", "question_compact_sha256": "52144ae39f0abe7a2ec02ea1690c5c8584445ea9dc3ca4154bec7be80b9a3ea0", "english_row_compact_sha256": "d6ba7da752a3d70c568203c1c281cf07450ac95c316f913bfbe17bd2366c81ca", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261028-r06-random-variable-independence-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261028-r06-random-variable-independence-foundations.en.jsonl" }, "mko_bindings": { "mko-random-variable-independence": { "sha256": "e8f6e97a13cd93c4cf37384ba6ab187d3ad4ca22c8f56260ed3707df0b8cc94e", "english_sha256": "d30f20ac634c6f78b55166d2cdea3b6fa5631c304d8ecd617b39b0a5aae747e8", "original_path": "public/data/mko/random-variable-independence.json", "english_path": "public/data/i18n/editions/mko-random-variable-independence.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a8b6a7c538e8e95c336dfae507795cf07012b597b3dd6104fa3ecad599076f77", "original_candidate_sha256": "84453f1c6496a7de67cedf14ad2e879cf6342bca567bb5f2304ac8d4c9401ca9", "original_candidate_line_sha256": "84453f1c6496a7de67cedf14ad2e879cf6342bca567bb5f2304ac8d4c9401ca9" }, "historical_source_state": { "review_state": { "derivation": "candidate_authored_and_locally_checked", "source_question_state": "candidate", "independent_AI_or_human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN" }, "scope_note": { "zh-Hant": "來源綁定教學推導候選；精確核對不等於獨立AI／人工審查或一般形式證明。原題、答案、MKO、公式、AST與歷史不變。", "en": "Source-bound pedagogical derivation candidate.Exact checking is not independent AI/human review or general formal proof.Original questions,answers,MKOs,formulas,ASTs and history remain unchanged." }, "label": "Frozen producer metadata; current per-question review is separate below." }, "revision": { "id": "review-20261026-007-p28", "review_date": "2026-10-26", "owner_authorized_reader_publication": true, "scope": "One hundred sixty family prototypes and their mathematical/bilingual reasoning variants, stems, boundary statements and every distinct explanation shape were reviewed in a context independent of the producer; parameter substitutions, source bindings, full math models, both answer strings and every arithmetic step were independently reconstructed per question from the original parameters only, with negative controls. This grouped method does not claim a separate literal AI reading of every repeated sentence or an extra AI voter.", "review_receipts": { "semantic_review_sha256": "e49bf5eaa968c09ebac4ad538ce58251671ed922c957cee1471cfc7b68934bfe", "question_first_witness_sha256": "a7976057c26051fab92342fc95d48af224466c0e1e4ab6ae4c8811c02772e6da" }, "status": "second_context_reviewed", "mathematical_semantics": true, "bilingual_semantics": true, "steps_reviewed": 13, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9b9f82d0087b84ed486c6e03789e8c213299122a1ff4a0c78e150304c5610467", "arithmetic_steps": 9 }, "uses_answer_as_premise": false, "human_review": "NOT_PERFORMED", "formalization": "not_formalized", "Lean": "NOT_RUN", "Canonical_truth_promotion": false, "authority": "Owner-authorized reviewed reader publication; original question/candidate states remain historical and unchanged." } }