{ "schema_version": "ocme-reviewed-question-reader-v1", "question_id": "q-20261014-advance-20261029-r06-iid-sequence-0041", "pilot_number": "R0027-14491", "number": 14491, "family_id": "qf-mod29-iid-identical-marginal", "title": { "zh-Hant": "第 14491 題：獨立同分布序列", "en": "Question 14491: Identical Distribution Means the Complete Marginal, Not Just Its Mean" }, "original_question": { "schema_version": "ocme-question-v0.1", "id": "q-20261014-advance-20261029-r06-iid-sequence-0041", "batch_id": "batch-20261014-advance-20261029-r06-iid-sequence-foundations", "language": "zh-Hant", "source_mko_ids": [ "mko-iid-sequence" ], "title_zh": "獨立同分布序列", "stem_zh": "IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[16,29],\"left_index\":15,\"right_index\":32}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。", "answer": { "kind": "expression", "value": "(15, 32, [0, 13/29], [1, 16/29], 是, full_Bernoulli_marginal_not_mean_only)", "explanation_zh": "Wave4第三層核對=(15, 32, [0, 13/29], [1, 16/29], 是, full_Bernoulli_marginal_not_mean_only)；依完整明示資料重算；完整有限／symbolic certificate不自動升格為未聲明的一般定理；無限序列使用symbolic product-law cylinder證書；相同期望pairwise或固定prefix不足且不推出LLN/CLT。" }, "difficulty": { "band": "undergraduate_core", "level": 2, "skills": [ "獨立同分布序列" ] }, "variant": { "family_id": "qf-mod29-iid-identical-marginal", "seed": 202610296041, "parameters": { "p": [ 16, 29 ], "left_index": 15, "right_index": 32, "domain": "Symbolic_Bernoulli_product_law", "operation": "iid-identical-marginal", "semantic_template": "qf-mod29-iid-identical-marginal/{\"left_index\":15,\"p\":[16,29],\"right_index\":32}" } }, "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、一般域／UFD、Hilbert、一般條件期望、LLN／CLT、商集、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-20261029-r06-iid-sequence-0041", "batch_id": "batch-20261014-advance-20261029-r06-iid-sequence-foundations", "locale": "en", "translation_state": "ai_candidate", "source": { "question_sha256": "b68b452eab09cc14cb760d12543e762ae117e977f2f4571659a834c019cbd6f9", "family_id": "qf-mod29-iid-identical-marginal", "seed": 202610296041, "semantic_template": "qf-mod29-iid-identical-marginal/{\"left_index\":15,\"p\":[16,29],\"right_index\":32}" }, "fields": { "title": "Identical Distribution Means the Complete Marginal, Not Just Its Mean", "stem": "Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=15, right_index=32. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only).", "answer": { "kind": "expression", "value": "(15, 32, [0, 13/29], [1, 16/29], Yes, full_Bernoulli_marginal_not_mean_only)", "explanation": "Both indices use mass 1−p at 0 and p at 1 under this same declared product law. Retain both outcome masses, including zeros; do not infer identical distribution from matching expectations alone. The exact result, retaining the requested order, is (15, 32, [0, 13/29], [1, 16/29], Yes, full_Bernoulli_marginal_not_mean_only)." }, "skills": [ "Identical Distribution Means the Complete Marginal, Not Just Its Mean" ], "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": "IID同時要求完整相同marginal與每個有限子族獨立。操作=iid-identical-marginal；資料={\"p\":[16,29],\"left_index\":15,\"right_index\":32}。請依 [left_index,right_index,zero_mass,one_mass,identical,boundary] 欄序回答。", "en": "Use the explicitly declared Bernoulli product-law model with exact p=16/29 in [0,1]. Infinite IID requires the complete identical Bernoulli marginal and mutual independence for every finite subfamily, represented here by the symbolic product-law cylinder certificate. Equal means, pairwise independence or checking one finite prefix is insufficient. Use exact reduced fractions with positive denominators, omitting /1 for integers. No law of large numbers, central limit theorem or additional Markov conclusion is asserted. left_index=15, right_index=32. Give (left_index,right_index,[0,zero_mass],[1,one_mass],identical?,full_Bernoulli_marginal_not_mean_only)." } } ], "steps": [ { "id": "S001", "uses": [ "G001" ], "before": "Bernoulli zero-outcome mass := sub(1, 16/29)", "after": "Bernoulli zero-outcome mass = 13/29", "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 Bernoulli zero-outcome mass，不以小數近似替代。", "en": "Recompute Bernoulli zero-outcome mass with exact rational arithmetic, not rounded decimals." } }, "arithmetic": { "operation": "sub", "inputs": [ "1", "16/29" ], "output": "13/29", "label": "Bernoulli zero-outcome mass" } }, { "id": "S002", "uses": [ "G001", "S001" ], "before": "raw Bernoulli parameter=[16, 29]", "after": "normalized p=16/29;mass at0=13/29;mass at1=16/29;symbolic product-law model explicitly declared", "relation": "implies", "rule_id": "Bernoulli_complete_marginal_and_product_law_premise", "conditions": [], "explanation": { "zh-Hant": "保留完整{0,1}邊際與宣告的乘積法則前提；不是從一組數據、相同期望或pairwise檢查猜無限IID。", "en": "Retain the complete{0,1} marginal and the explicitly declared product-law premise.Infinite IID is not inferred from a data set,equal means or pairwise checks." }, "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": "indices=[15, 32];same declared Bernoulli product law", "after": "both complete marginals:mass0=13/29;mass1=16/29;identical=是", "relation": "implies", "rule_id": "IID_full_identical_marginal_not_general_mean_test", "conditions": [], "explanation": { "zh-Hant": "對兩個索引都列出0與1兩點質量，包含退化參數的零質量；一般變數只比期望相同不能證同分布，更不能證獨立。", "en": "List both outcome masses at each index,including zero masses for degenerate parameters.Equal means alone do not establish identical distribution for general variables,let alone independence." }, "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": [ 15, 32, [ "0", "13/29" ], [ "1", "16/29" ], true, "full_Bernoulli_marginal_not_mean_only" ] }, "conclusion": { "canonical_value": "(15, 32, [0, 13/29], [1, 16/29], 是, full_Bernoulli_marginal_not_mean_only)", "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-20261029-r06-iid-sequence-foundations", "batch_path": "public/data/questions/batches/batch-20261014-advance-20261029-r06-iid-sequence-foundations/questions.jsonl", "batch_sha256": "eb1e6ee54e11c0710a74de69c6e79201e0d20149f0261f850a24316a257e8115", "original_line": 41, "original_line_sha256": "7f2ff8aa2330a47628c3381f6c0f7353589c086e75031d5ade90f49a79a22590", "question_compact_sha256": "b68b452eab09cc14cb760d12543e762ae117e977f2f4571659a834c019cbd6f9", "english_row_compact_sha256": "8f71123f68209fdbd6d799a0a6a67da137ddb22830a385c4d5f2ab1aeaf93d3b", "english_manifest_path": "public/data/i18n/editions/batch-20261014-advance-20261029-r06-iid-sequence-foundations.en.json", "english_rows_path": "public/data/i18n/questions/batch-20261014-advance-20261029-r06-iid-sequence-foundations.en.jsonl" }, "mko_bindings": { "mko-iid-sequence": { "sha256": "bac79da4634be201a10f08ec4489946a636db5ebc2e628c0b1830b743f4ddbd0", "english_sha256": "9fc20d5d2102056dfc01584c16ff07f2a621a4acee305e717a7c68822a5dfa48", "original_path": "public/data/mko/iid-sequence.json", "english_path": "public/data/i18n/editions/mko-iid-sequence.en.json" } }, "selected_manifest_sha256": "5233edba9654a6cffd8b6b2d83c932f82ded15f9c0b7c733bd049943e672acff", "selected_part_sha256": "a8b6a7c538e8e95c336dfae507795cf07012b597b3dd6104fa3ecad599076f77", "original_candidate_sha256": "282bd96206983a152c77397a554eb79958272c9d504f9ef4a2b35183263453ca", "original_candidate_line_sha256": "282bd96206983a152c77397a554eb79958272c9d504f9ef4a2b35183263453ca" }, "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": 3, "method": "family_template_semantic_review_plus_independent_per_instance_reconstruction", "exact_question_first_check": { "model_sha256": "9ec94fb5b47cf2c568479c7d82ab65ce3cd048124b56ac301b4649e8fe2a6536", "arithmetic_steps": 1 }, "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." } }