{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0001","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[1, 0, 2]、D=3，權重為質量/D；X值=[-3, 99, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-1/1, 0/1, 0/1]","explanation_zh":"逐點期望項=[-1/1, 0/1, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105001,"parameters":{"masses":[1,0,2],"D":3,"values":[-3,99,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-3,\"1/3\"],[\"zero\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0002","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 1]、D=2，權重為質量/D；X值=[0, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[0/1, 0/1]","explanation_zh":"逐點期望項=[0/1, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105002,"parameters":{"masses":[1,1],"D":2,"values":[0,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[\"zero\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0003","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[3, 2, 1]、D=6，權重為質量/D；X值=[-2, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-1/1, 2/3, 1/6]","explanation_zh":"逐點期望項=[-1/1, 2/3, 1/6]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105003,"parameters":{"masses":[3,2,1],"D":6,"values":[-2,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-2,\"1/2\"],[2,\"1/3\"],[1,\"1/6\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0004","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[4, 2, 1]、D=7，權重為質量/D；X值=[-3, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-12/7, 4/7, -1/7]","explanation_zh":"逐點期望項=[-12/7, 4/7, -1/7]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105004,"parameters":{"masses":[4,2,1],"D":7,"values":[-3,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-3,\"4/7\"],[2,\"2/7\"],[-1,\"1/7\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0005","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[5, 2, 1]、D=8，權重為質量/D；X值=[-4, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-5/2, 1/2, 0/1]","explanation_zh":"逐點期望項=[-5/2, 1/2, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105005,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-4,\"5/8\"],[2,\"1/4\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0006","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[6, 2, 1]、D=9，權重為質量/D；X值=[-5, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-10/3, 4/9, 1/9]","explanation_zh":"逐點期望項=[-10/3, 4/9, 1/9]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105006,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-5,\"2/3\"],[2,\"2/9\"],[1,\"1/9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0007","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[7, 2, 1]、D=10，權重為質量/D；X值=[-6, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-21/5, 2/5, -1/10]","explanation_zh":"逐點期望項=[-21/5, 2/5, -1/10]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105007,"parameters":{"masses":[7,2,1],"D":10,"values":[-6,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-6,\"7/10\"],[2,\"1/5\"],[-1,\"1/10\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0008","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[8, 2, 1]、D=11，權重為質量/D；X值=[-7, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-56/11, 4/11, 0/1]","explanation_zh":"逐點期望項=[-56/11, 4/11, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105008,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-7,\"8/11\"],[2,\"2/11\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0009","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[9, 2, 1]、D=12，權重為質量/D；X值=[-8, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-6/1, 1/3, 1/12]","explanation_zh":"逐點期望項=[-6/1, 1/3, 1/12]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105009,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-8,\"3/4\"],[2,\"1/6\"],[1,\"1/12\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0010","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[10, 2, 1]、D=13，權重為質量/D；X值=[-9, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-90/13, 4/13, -1/13]","explanation_zh":"逐點期望項=[-90/13, 4/13, -1/13]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105010,"parameters":{"masses":[10,2,1],"D":13,"values":[-9,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-9,\"10/13\"],[2,\"2/13\"],[-1,\"1/13\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0011","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[11, 2, 1]、D=14，權重為質量/D；X值=[-10, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-55/7, 2/7, 0/1]","explanation_zh":"逐點期望項=[-55/7, 2/7, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105011,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-10,\"11/14\"],[2,\"1/7\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0012","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 2, 1]、D=15，權重為質量/D；X值=[-11, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-44/5, 4/15, 1/15]","explanation_zh":"逐點期望項=[-44/5, 4/15, 1/15]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105012,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-11,\"4/5\"],[2,\"2/15\"],[1,\"1/15\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0013","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[13, 2, 1]、D=16，權重為質量/D；X值=[-12, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-39/4, 1/4, -1/16]","explanation_zh":"逐點期望項=[-39/4, 1/4, -1/16]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105013,"parameters":{"masses":[13,2,1],"D":16,"values":[-12,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-12,\"13/16\"],[2,\"1/8\"],[-1,\"1/16\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0014","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[14, 2, 1]、D=17，權重為質量/D；X值=[-13, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-182/17, 4/17, 0/1]","explanation_zh":"逐點期望項=[-182/17, 4/17, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105014,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-13,\"14/17\"],[2,\"2/17\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0015","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[15, 2, 1]、D=18，權重為質量/D；X值=[-14, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-35/3, 2/9, 1/18]","explanation_zh":"逐點期望項=[-35/3, 2/9, 1/18]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105015,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-14,\"5/6\"],[2,\"1/9\"],[1,\"1/18\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0016","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[16, 2, 1]、D=19，權重為質量/D；X值=[-15, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-240/19, 4/19, -1/19]","explanation_zh":"逐點期望項=[-240/19, 4/19, -1/19]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105016,"parameters":{"masses":[16,2,1],"D":19,"values":[-15,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-15,\"16/19\"],[2,\"2/19\"],[-1,\"1/19\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0017","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[17, 2, 1]、D=20，權重為質量/D；X值=[-16, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-68/5, 1/5, 0/1]","explanation_zh":"逐點期望項=[-68/5, 1/5, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105017,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-16,\"17/20\"],[2,\"1/10\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0018","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[18, 2, 1]、D=21，權重為質量/D；X值=[-17, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-102/7, 4/21, 1/21]","explanation_zh":"逐點期望項=[-102/7, 4/21, 1/21]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105018,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-17,\"6/7\"],[2,\"2/21\"],[1,\"1/21\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0019","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[19, 2, 1]、D=22，權重為質量/D；X值=[-18, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-171/11, 2/11, -1/22]","explanation_zh":"逐點期望項=[-171/11, 2/11, -1/22]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105019,"parameters":{"masses":[19,2,1],"D":22,"values":[-18,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-18,\"19/22\"],[2,\"1/11\"],[-1,\"1/22\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0020","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[20, 2, 1]、D=23，權重為質量/D；X值=[-19, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-380/23, 4/23, 0/1]","explanation_zh":"逐點期望項=[-380/23, 4/23, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105020,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-19,\"20/23\"],[2,\"2/23\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0021","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 2, 1]、D=24，權重為質量/D；X值=[-20, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-35/2, 1/6, 1/24]","explanation_zh":"逐點期望項=[-35/2, 1/6, 1/24]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105021,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-20,\"7/8\"],[2,\"1/12\"],[1,\"1/24\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0022","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[22, 2, 1]、D=25，權重為質量/D；X值=[-21, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-462/25, 4/25, -1/25]","explanation_zh":"逐點期望項=[-462/25, 4/25, -1/25]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105022,"parameters":{"masses":[22,2,1],"D":25,"values":[-21,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-21,\"22/25\"],[2,\"2/25\"],[-1,\"1/25\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0023","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[23, 2, 1]、D=26，權重為質量/D；X值=[-22, 2, 0]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-253/13, 2/13, 0/1]","explanation_zh":"逐點期望項=[-253/13, 2/13, 0/1]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105023,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,2,0],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-22,\"23/26\"],[2,\"1/13\"],[\"zero\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0024","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[24, 2, 1]、D=27，權重為質量/D；X值=[-23, 2, 1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-184/9, 4/27, 1/27]","explanation_zh":"逐點期望項=[-184/9, 4/27, 1/27]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105024,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,2,1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-23,\"8/9\"],[2,\"2/27\"],[1,\"1/27\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0025","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"逐點期望貢獻","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[25, 2, 1]、D=28，權重為質量/D；X值=[-24, 2, -1]。按索引0起列每項w_i X_i，保留零項，以最簡分數表示。","answer":{"kind":"expression","value":"[-150/7, 1/7, -1/28]","explanation_zh":"逐點期望項=[-150/7, 1/7, -1/28]。逐項以質量乘值再除D；零權重異值貢獻零，負值可以給負貢獻。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點期望貢獻"]},"variant":{"family_id":"qf-density10-expectation-contributions","seed":202610105025,"parameters":{"masses":[25,2,1],"D":28,"values":[-24,2,-1],"operation":"density10_expectation_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-contributions/[[-24,\"25/28\"],[2,\"1/14\"],[-1,\"1/28\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0026","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[1, 0, 2]、D=3，權重為質量/D；X值=[4, 99, 4]、Y值=[4, -50, 4]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"0/1","explanation_zh":"共模型差期望=0/1。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105026,"parameters":{"masses":[1,0,2],"D":3,"values":[4,99,4],"other":[4,-50,4],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"0\",\"1/1\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0027","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 1]、D=2，權重為質量/D；X值=[0, 2]、Y值=[2, 0]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"0/1","explanation_zh":"共模型差期望=0/1。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105027,"parameters":{"masses":[1,1],"D":2,"values":[0,2],"other":[2,0],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-2\",\"1/2\"],[\"2\",\"1/2\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0028","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[3, 2, 1]、D=6，權重為質量/D；X值=[-2, 2, 3]、Y值=[1, -1, -2]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"1/3","explanation_zh":"共模型差期望=1/3。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105028,"parameters":{"masses":[3,2,1],"D":6,"values":[-2,2,3],"other":[1,-1,-2],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-3\",\"1/2\"],[\"3\",\"1/3\"],[\"5\",\"1/6\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0029","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[4, 2, 1]、D=7，權重為質量/D；X值=[-3, 2, 4]、Y值=[1, -1, -3]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-3/7","explanation_zh":"共模型差期望=-3/7。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105029,"parameters":{"masses":[4,2,1],"D":7,"values":[-3,2,4],"other":[1,-1,-3],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-4\",\"4/7\"],[\"3\",\"2/7\"],[\"7\",\"1/7\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0030","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[5, 2, 1]、D=8，權重為質量/D；X值=[-4, 2, 5]、Y值=[1, -1, -4]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-5/4","explanation_zh":"共模型差期望=-5/4。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105030,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,2,5],"other":[1,-1,-4],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-5\",\"5/8\"],[\"3\",\"1/4\"],[\"9\",\"1/8\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0031","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[6, 2, 1]、D=9，權重為質量/D；X值=[-5, 2, 6]、Y值=[1, -1, -5]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-19/9","explanation_zh":"共模型差期望=-19/9。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105031,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,2,6],"other":[1,-1,-5],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-6\",\"2/3\"],[\"11\",\"1/9\"],[\"3\",\"2/9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0032","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[7, 2, 1]、D=10，權重為質量/D；X值=[-6, 2, 7]、Y值=[1, -1, -6]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"共模型差期望=-3/1。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105032,"parameters":{"masses":[7,2,1],"D":10,"values":[-6,2,7],"other":[1,-1,-6],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-7\",\"7/10\"],[\"13\",\"1/10\"],[\"3\",\"1/5\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0033","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[8, 2, 1]、D=11，權重為質量/D；X值=[-7, 2, 8]、Y值=[1, -1, -7]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-43/11","explanation_zh":"共模型差期望=-43/11。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105033,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,2,8],"other":[1,-1,-7],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-8\",\"8/11\"],[\"15\",\"1/11\"],[\"3\",\"2/11\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0034","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[9, 2, 1]、D=12，權重為質量/D；X值=[-8, 2, 9]、Y值=[1, -1, -8]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-29/6","explanation_zh":"共模型差期望=-29/6。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105034,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,2,9],"other":[1,-1,-8],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-9\",\"3/4\"],[\"17\",\"1/12\"],[\"3\",\"1/6\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0035","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[10, 2, 1]、D=13，權重為質量/D；X值=[-9, 2, 10]、Y值=[1, -1, -9]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-75/13","explanation_zh":"共模型差期望=-75/13。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105035,"parameters":{"masses":[10,2,1],"D":13,"values":[-9,2,10],"other":[1,-1,-9],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-10\",\"10/13\"],[\"19\",\"1/13\"],[\"3\",\"2/13\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0036","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[11, 2, 1]、D=14，權重為質量/D；X值=[-10, 2, 11]、Y值=[1, -1, -10]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-47/7","explanation_zh":"共模型差期望=-47/7。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105036,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,2,11],"other":[1,-1,-10],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-11\",\"11/14\"],[\"21\",\"1/14\"],[\"3\",\"1/7\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0037","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 2, 1]、D=15，權重為質量/D；X值=[-11, 2, 12]、Y值=[1, -1, -11]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-23/3","explanation_zh":"共模型差期望=-23/3。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105037,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,2,12],"other":[1,-1,-11],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-12\",\"4/5\"],[\"23\",\"1/15\"],[\"3\",\"2/15\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0038","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[13, 2, 1]、D=16，權重為質量/D；X值=[-12, 2, 13]、Y值=[1, -1, -12]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-69/8","explanation_zh":"共模型差期望=-69/8。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105038,"parameters":{"masses":[13,2,1],"D":16,"values":[-12,2,13],"other":[1,-1,-12],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-13\",\"13/16\"],[\"25\",\"1/16\"],[\"3\",\"1/8\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0039","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[14, 2, 1]、D=17，權重為質量/D；X值=[-13, 2, 14]、Y值=[1, -1, -13]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-163/17","explanation_zh":"共模型差期望=-163/17。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105039,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,2,14],"other":[1,-1,-13],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-14\",\"14/17\"],[\"27\",\"1/17\"],[\"3\",\"2/17\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0040","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[15, 2, 1]、D=18，權重為質量/D；X值=[-14, 2, 15]、Y值=[1, -1, -14]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-95/9","explanation_zh":"共模型差期望=-95/9。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105040,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,2,15],"other":[1,-1,-14],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-15\",\"5/6\"],[\"29\",\"1/18\"],[\"3\",\"1/9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0041","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[16, 2, 1]、D=19，權重為質量/D；X值=[-15, 2, 16]、Y值=[1, -1, -15]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-219/19","explanation_zh":"共模型差期望=-219/19。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105041,"parameters":{"masses":[16,2,1],"D":19,"values":[-15,2,16],"other":[1,-1,-15],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-16\",\"16/19\"],[\"3\",\"2/19\"],[\"31\",\"1/19\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0042","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[17, 2, 1]、D=20，權重為質量/D；X值=[-16, 2, 17]、Y值=[1, -1, -16]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-25/2","explanation_zh":"共模型差期望=-25/2。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105042,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,2,17],"other":[1,-1,-16],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-17\",\"17/20\"],[\"3\",\"1/10\"],[\"33\",\"1/20\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0043","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[18, 2, 1]、D=21，權重為質量/D；X值=[-17, 2, 18]、Y值=[1, -1, -17]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-283/21","explanation_zh":"共模型差期望=-283/21。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105043,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,2,18],"other":[1,-1,-17],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-18\",\"6/7\"],[\"3\",\"2/21\"],[\"35\",\"1/21\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0044","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[19, 2, 1]、D=22，權重為質量/D；X值=[-18, 2, 19]、Y值=[1, -1, -18]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-159/11","explanation_zh":"共模型差期望=-159/11。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105044,"parameters":{"masses":[19,2,1],"D":22,"values":[-18,2,19],"other":[1,-1,-18],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-19\",\"19/22\"],[\"3\",\"1/11\"],[\"37\",\"1/22\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0045","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[20, 2, 1]、D=23，權重為質量/D；X值=[-19, 2, 20]、Y值=[1, -1, -19]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-355/23","explanation_zh":"共模型差期望=-355/23。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105045,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,2,20],"other":[1,-1,-19],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-20\",\"20/23\"],[\"3\",\"2/23\"],[\"39\",\"1/23\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0046","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 2, 1]、D=24，權重為質量/D；X值=[-20, 2, 21]、Y值=[1, -1, -20]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-197/12","explanation_zh":"共模型差期望=-197/12。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105046,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,2,21],"other":[1,-1,-20],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-21\",\"7/8\"],[\"3\",\"1/12\"],[\"41\",\"1/24\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0047","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[22, 2, 1]、D=25，權重為質量/D；X值=[-21, 2, 22]、Y值=[1, -1, -21]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-87/5","explanation_zh":"共模型差期望=-87/5。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105047,"parameters":{"masses":[22,2,1],"D":25,"values":[-21,2,22],"other":[1,-1,-21],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-22\",\"22/25\"],[\"3\",\"2/25\"],[\"43\",\"1/25\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0048","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[23, 2, 1]、D=26，權重為質量/D；X值=[-22, 2, 23]、Y值=[1, -1, -22]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-239/13","explanation_zh":"共模型差期望=-239/13。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105048,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,2,23],"other":[1,-1,-22],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-23\",\"23/26\"],[\"3\",\"1/13\"],[\"45\",\"1/26\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0049","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[24, 2, 1]、D=27，權重為質量/D；X值=[-23, 2, 24]、Y值=[1, -1, -23]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-523/27","explanation_zh":"共模型差期望=-523/27。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105049,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,2,24],"other":[1,-1,-23],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-24\",\"8/9\"],[\"3\",\"2/27\"],[\"47\",\"1/27\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0050","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共模型的差期望","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[25, 2, 1]、D=28，權重為質量/D；X值=[-24, 2, 25]、Y值=[1, -1, -24]，X與Y共用S和權重，不假設獨立。求 E[X]−E[Y]，亦即 E[X−Y]。","answer":{"kind":"expression","value":"-285/14","explanation_zh":"共模型差期望=-285/14。分別加權求兩個均值再相減；共同模型線性計算不需要獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型的差期望"]},"variant":{"family_id":"qf-density10-expectation-difference","seed":202610105050,"parameters":{"masses":[25,2,1],"D":28,"values":[-24,2,25],"other":[1,-1,-24],"operation":"density10_expectation_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-difference/[[\"-25\",\"25/28\"],[\"3\",\"1/14\"],[\"49\",\"1/28\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0051","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 1]、D=2，權重為質量/D；X值=[0, 1]、Y值=[0, 1]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(1/2, 1/4)","explanation_zh":"乘積期望與均值積=(1/2, 1/4)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105051,"parameters":{"masses":[1,1],"D":2,"values":[0,1],"other":[0,1],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[0,0,\"1/2\"],[1,1,\"1/2\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0052","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,1}，F=𝒫(S)，質量=[1, 2]、D=3，權重為質量/D；X值=[3, 3]、Y值=[-1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(3/1, 3/1)","explanation_zh":"乘積期望與均值積=(3/1, 3/1)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105052,"parameters":{"masses":[1,2],"D":3,"values":[3,3],"other":[-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,3,\"1/3\"],[2,3,\"2/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0053","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[1, 0, 2]、D=3，權重為質量/D；X值=[-2, 99, 1]、Y值=[1, -100, -2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-2/1, 0/1)","explanation_zh":"乘積期望與均值積=(-2/1, 0/1)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105053,"parameters":{"masses":[1,0,2],"D":3,"values":[-2,99,1],"other":[1,-100,-2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-2,1,\"1/3\"],[1,-2,\"2/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0054","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[4, 2, 1]、D=7，權重為質量/D；X值=[-3, 2, 4]、Y值=[5, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-8/1, -80/49)","explanation_zh":"乘積期望與均值積=(-8/1, -80/49)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105054,"parameters":{"masses":[4,2,1],"D":7,"values":[-3,2,4],"other":[5,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/7\"],[2,4,\"1/7\"],[5,-3,\"4/7\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0055","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[5, 2, 1]、D=8，權重為質量/D；X值=[-4, 2, 5]、Y值=[6, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-57/4, -165/32)","explanation_zh":"乘積期望與均值積=(-57/4, -165/32)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105055,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,2,5],"other":[6,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/4\"],[2,5,\"1/8\"],[6,-4,\"5/8\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0056","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[6, 2, 1]、D=9，權重為質量/D；X值=[-5, 2, 6]、Y值=[7, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-202/9, -280/27)","explanation_zh":"乘積期望與均值積=(-202/9, -280/27)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105056,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,2,6],"other":[7,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/9\"],[2,6,\"1/9\"],[7,-5,\"2/3\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0057","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[7, 2, 1]、D=10，權重為質量/D；X值=[-6, 2, 7]、Y值=[8, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-163/5, -434/25)","explanation_zh":"乘積期望與均值積=(-163/5, -434/25)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105057,"parameters":{"masses":[7,2,1],"D":10,"values":[-6,2,7],"other":[8,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/5\"],[2,7,\"1/10\"],[8,-6,\"7/10\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0058","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[8, 2, 1]、D=11，權重為質量/D；X值=[-7, 2, 8]、Y值=[9, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-492/11, -288/11)","explanation_zh":"乘積期望與均值積=(-492/11, -288/11)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105058,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,2,8],"other":[9,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/11\"],[2,8,\"1/11\"],[9,-7,\"8/11\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0059","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[9, 2, 1]、D=12，權重為質量/D；X值=[-8, 2, 9]、Y值=[10, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-353/6, -295/8)","explanation_zh":"乘積期望與均值積=(-353/6, -295/8)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105059,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,2,9],"other":[10,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/6\"],[10,-8,\"3/4\"],[2,9,\"1/12\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0060","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[10, 2, 1]、D=13，權重為質量/D；X值=[-9, 2, 10]、Y值=[11, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-974/13, -8360/169)","explanation_zh":"乘積期望與均值積=(-974/13, -8360/169)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105060,"parameters":{"masses":[10,2,1],"D":13,"values":[-9,2,10],"other":[11,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/13\"],[11,-9,\"10/13\"],[2,10,\"1/13\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0061","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[11, 2, 1]、D=14，權重為質量/D；X值=[-10, 2, 11]、Y值=[12, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-93/1, -3135/49)","explanation_zh":"乘積期望與均值積=(-93/1, -3135/49)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105061,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,2,11],"other":[12,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/7\"],[12,-10,\"11/14\"],[2,11,\"1/14\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0062","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 2, 1]、D=15，權重為質量/D；X值=[-11, 2, 12]、Y值=[13, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-1696/15, -6032/75)","explanation_zh":"乘積期望與均值積=(-1696/15, -6032/75)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105062,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,2,12],"other":[13,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/15\"],[13,-11,\"4/5\"],[2,12,\"1/15\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0063","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[13, 2, 1]、D=16，權重為質量/D；X值=[-12, 2, 13]、Y值=[14, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-1081/8, -12649/128)","explanation_zh":"乘積期望與均值積=(-1081/8, -12649/128)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105063,"parameters":{"masses":[13,2,1],"D":16,"values":[-12,2,13],"other":[14,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/8\"],[14,-12,\"13/16\"],[2,13,\"1/16\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0064","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[14, 2, 1]、D=17，權重為質量/D；X值=[-13, 2, 14]、Y值=[15, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-2706/17, -34440/289)","explanation_zh":"乘積期望與均值積=(-2706/17, -34440/289)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105064,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,2,14],"other":[15,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/17\"],[15,-13,\"14/17\"],[2,14,\"1/17\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0065","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[15, 2, 1]、D=18，權重為質量/D；X值=[-14, 2, 15]、Y值=[16, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-1667/9, -3820/27)","explanation_zh":"乘積期望與均值積=(-1667/9, -3820/27)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105065,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,2,15],"other":[16,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/9\"],[16,-14,\"5/6\"],[2,15,\"1/18\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0066","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[16, 2, 1]、D=19，權重為質量/D；X值=[-15, 2, 16]、Y值=[17, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-4052/19, -59840/361)","explanation_zh":"乘積期望與均值積=(-4052/19, -59840/361)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105066,"parameters":{"masses":[16,2,1],"D":19,"values":[-15,2,16],"other":[17,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/19\"],[17,-15,\"16/19\"],[2,16,\"1/19\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0067","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[17, 2, 1]、D=20，權重為質量/D；X值=[-16, 2, 17]、Y值=[18, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-2433/10, -38403/200)","explanation_zh":"乘積期望與均值積=(-2433/10, -38403/200)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105067,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,2,17],"other":[18,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/10\"],[18,-16,\"17/20\"],[2,17,\"1/20\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0068","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[18, 2, 1]、D=21，權重為質量/D；X值=[-17, 2, 18]、Y值=[19, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-826/3, -10792/49)","explanation_zh":"乘積期望與均值積=(-826/3, -10792/49)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105068,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,2,18],"other":[19,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/21\"],[19,-17,\"6/7\"],[2,18,\"1/21\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0069","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[19, 2, 1]、D=22，權重為質量/D；X值=[-18, 2, 19]、Y值=[20, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-3403/11, -2755/11)","explanation_zh":"乘積期望與均值積=(-3403/11, -2755/11)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105069,"parameters":{"masses":[19,2,1],"D":22,"values":[-18,2,19],"other":[20,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/11\"],[2,19,\"1/22\"],[20,-18,\"19/22\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0070","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[20, 2, 1]、D=23，權重為質量/D；X值=[-19, 2, 20]、Y值=[21, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-7944/23, -149520/529)","explanation_zh":"乘積期望與均值積=(-7944/23, -149520/529)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105070,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,2,20],"other":[21,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/23\"],[2,20,\"1/23\"],[21,-19,\"20/23\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0071","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 2, 1]、D=24，權重為質量/D；X值=[-20, 2, 21]、Y值=[22, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-4601/12, -30415/96)","explanation_zh":"乘積期望與均值積=(-4601/12, -30415/96)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105071,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,2,21],"other":[22,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/12\"],[2,21,\"1/24\"],[22,-20,\"7/8\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0072","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[22, 2, 1]、D=25，權重為質量/D；X值=[-21, 2, 22]、Y值=[23, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-10586/25, -220616/625)","explanation_zh":"乘積期望與均值積=(-10586/25, -220616/625)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105072,"parameters":{"masses":[22,2,1],"D":25,"values":[-21,2,22],"other":[23,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/25\"],[2,22,\"1/25\"],[23,-21,\"22/25\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0073","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[23, 2, 1]、D=26，權重為質量/D；X值=[-22, 2, 23]、Y值=[24, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-6051/13, -66102/169)","explanation_zh":"乘積期望與均值積=(-6051/13, -66102/169)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105073,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,2,23],"other":[24,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/13\"],[2,23,\"1/26\"],[24,-22,\"23/26\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0074","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[24, 2, 1]、D=27，權重為質量/D；X值=[-23, 2, 24]、Y值=[25, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-13756/27, -104800/243)","explanation_zh":"乘積期望與均值積=(-13756/27, -104800/243)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105074,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,2,24],"other":[25,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/27\"],[2,24,\"1/27\"],[25,-23,\"8/9\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0075","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"乘積期望與均值積的有序對","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[25, 2, 1]、D=28，權重為質量/D；X值=[-24, 2, 25]、Y值=[26, -1, 2]，X與Y共用S和權重，不假設獨立。依序求 (E[XY], E[X]E[Y])。","answer":{"kind":"expression","value":"(-1111/2, -185575/392)","explanation_zh":"乘積期望與均值積=(-1111/2, -185575/392)。逐點乘積後加權，另算兩均值的積；數值相等也不推出獨立。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["乘積期望與均值積的有序對"]},"variant":{"family_id":"qf-density10-expectation-product-pair","seed":202610105075,"parameters":{"masses":[25,2,1],"D":28,"values":[-24,2,25],"other":[26,-1,2],"operation":"density10_expectation_product_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/14\"],[2,25,\"1/28\"],[26,-24,\"25/28\"]]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0076","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[1, 0, 2]、D=3，權重為質量/D；X值=[null, 99, 3]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=1/1。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"未知整數值=-3/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105076,"parameters":{"masses":[1,0,2],"D":3,"values":[null,99,3],"k":0,"mean_n":1,"mean_d":1,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[\"zero\"],[3,\"2/3\"]],\"1/3\",\"1/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0077","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,0}，F=𝒫(S)，質量=[1]、D=1，權重為質量/D；X值=[null]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=0/1。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"0/1","explanation_zh":"未知整數值=0/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105077,"parameters":{"masses":[1],"D":1,"values":[null],"k":0,"mean_n":0,"mean_d":1,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null],\"1/1\",\"0/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0078","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[3, 2, 1]、D=6，權重為質量/D；X值=[-2, 4, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=1/3。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"0/1","explanation_zh":"未知整數值=0/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105078,"parameters":{"masses":[3,2,1],"D":6,"values":[-2,4,null],"k":2,"mean_n":1,"mean_d":3,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-2,\"1/2\"],[4,\"1/3\"],null],\"1/6\",\"1/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0079","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[4, 2, 1]、D=7，權重為質量/D；X值=[null, 5, -1]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-3/7。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"未知整數值=-3/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105079,"parameters":{"masses":[4,2,1],"D":7,"values":[null,5,-1],"k":0,"mean_n":-3,"mean_d":7,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[5,\"2/7\"],[-1,\"1/7\"]],\"4/7\",\"-3/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0080","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[5, 2, 1]、D=8，權重為質量/D；X值=[-4, null, -2]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-5/4。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"6/1","explanation_zh":"未知整數值=6/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105080,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,null,-2],"k":1,"mean_n":-5,"mean_d":4,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-4,\"5/8\"],null,[-2,\"1/8\"]],\"1/4\",\"-5/4\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0081","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[6, 2, 1]、D=9，權重為質量/D；X值=[-5, 7, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-19/9。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"未知整數值=-3/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105081,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,7,null],"k":2,"mean_n":-19,"mean_d":9,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-5,\"2/3\"],[7,\"2/9\"],null],\"1/9\",\"-19/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0082","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[7, 2, 1]、D=10，權重為質量/D；X值=[null, 8, -4]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-3/1。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-6/1","explanation_zh":"未知整數值=-6/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105082,"parameters":{"masses":[7,2,1],"D":10,"values":[null,8,-4],"k":0,"mean_n":-3,"mean_d":1,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[8,\"1/5\"],[-4,\"1/10\"]],\"7/10\",\"-3/1\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0083","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[8, 2, 1]、D=11，權重為質量/D；X值=[-7, null, -5]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-43/11。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"9/1","explanation_zh":"未知整數值=9/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105083,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,null,-5],"k":1,"mean_n":-43,"mean_d":11,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-7,\"8/11\"],null,[-5,\"1/11\"]],\"2/11\",\"-43/11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0084","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[9, 2, 1]、D=12，權重為質量/D；X值=[-8, 10, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-29/6。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-6/1","explanation_zh":"未知整數值=-6/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105084,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,10,null],"k":2,"mean_n":-29,"mean_d":6,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-8,\"3/4\"],[10,\"1/6\"],null],\"1/12\",\"-29/6\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0085","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[10, 2, 1]、D=13，權重為質量/D；X值=[null, 11, -7]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-75/13。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"未知整數值=-9/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105085,"parameters":{"masses":[10,2,1],"D":13,"values":[null,11,-7],"k":0,"mean_n":-75,"mean_d":13,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[11,\"2/13\"],[-7,\"1/13\"]],\"10/13\",\"-75/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0086","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[11, 2, 1]、D=14，權重為質量/D；X值=[-10, null, -8]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-47/7。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"12/1","explanation_zh":"未知整數值=12/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105086,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,null,-8],"k":1,"mean_n":-47,"mean_d":7,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-10,\"11/14\"],null,[-8,\"1/14\"]],\"1/7\",\"-47/7\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0087","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[12, 2, 1]、D=15，權重為質量/D；X值=[-11, 13, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-23/3。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"未知整數值=-9/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105087,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,13,null],"k":2,"mean_n":-23,"mean_d":3,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-11,\"4/5\"],[13,\"2/15\"],null],\"1/15\",\"-23/3\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0088","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[13, 2, 1]、D=16，權重為質量/D；X值=[null, 14, -10]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-69/8。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-12/1","explanation_zh":"未知整數值=-12/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105088,"parameters":{"masses":[13,2,1],"D":16,"values":[null,14,-10],"k":0,"mean_n":-69,"mean_d":8,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[14,\"1/8\"],[-10,\"1/16\"]],\"13/16\",\"-69/8\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0089","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[14, 2, 1]、D=17，權重為質量/D；X值=[-13, null, -11]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-163/17。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"15/1","explanation_zh":"未知整數值=15/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105089,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,null,-11],"k":1,"mean_n":-163,"mean_d":17,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-13,\"14/17\"],null,[-11,\"1/17\"]],\"2/17\",\"-163/17\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0090","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[15, 2, 1]、D=18，權重為質量/D；X值=[-14, 16, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-95/9。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-12/1","explanation_zh":"未知整數值=-12/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105090,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,16,null],"k":2,"mean_n":-95,"mean_d":9,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-14,\"5/6\"],[16,\"1/9\"],null],\"1/18\",\"-95/9\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0091","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[16, 2, 1]、D=19，權重為質量/D；X值=[null, 17, -13]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-219/19。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-15/1","explanation_zh":"未知整數值=-15/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105091,"parameters":{"masses":[16,2,1],"D":19,"values":[null,17,-13],"k":0,"mean_n":-219,"mean_d":19,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[17,\"2/19\"],[-13,\"1/19\"]],\"16/19\",\"-219/19\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0092","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[17, 2, 1]、D=20，權重為質量/D；X值=[-16, null, -14]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-25/2。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"18/1","explanation_zh":"未知整數值=18/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105092,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,null,-14],"k":1,"mean_n":-25,"mean_d":2,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-16,\"17/20\"],null,[-14,\"1/20\"]],\"1/10\",\"-25/2\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0093","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[18, 2, 1]、D=21，權重為質量/D；X值=[-17, 19, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-283/21。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-15/1","explanation_zh":"未知整數值=-15/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105093,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,19,null],"k":2,"mean_n":-283,"mean_d":21,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-17,\"6/7\"],[19,\"2/21\"],null],\"1/21\",\"-283/21\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0094","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[19, 2, 1]、D=22，權重為質量/D；X值=[null, 20, -16]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-159/11。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-18/1","explanation_zh":"未知整數值=-18/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105094,"parameters":{"masses":[19,2,1],"D":22,"values":[null,20,-16],"k":0,"mean_n":-159,"mean_d":11,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[20,\"1/11\"],[-16,\"1/22\"]],\"19/22\",\"-159/11\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0095","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[20, 2, 1]、D=23，權重為質量/D；X值=[-19, null, -17]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-355/23。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"21/1","explanation_zh":"未知整數值=21/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105095,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,null,-17],"k":1,"mean_n":-355,"mean_d":23,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-19,\"20/23\"],null,[-17,\"1/23\"]],\"2/23\",\"-355/23\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0096","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[21, 2, 1]、D=24，權重為質量/D；X值=[-20, 22, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-197/12。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-18/1","explanation_zh":"未知整數值=-18/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105096,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,22,null],"k":2,"mean_n":-197,"mean_d":12,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-20,\"7/8\"],[22,\"1/12\"],null],\"1/24\",\"-197/12\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0097","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[22, 2, 1]、D=25，權重為質量/D；X值=[null, 23, -19]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-87/5。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-21/1","explanation_zh":"未知整數值=-21/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105097,"parameters":{"masses":[22,2,1],"D":25,"values":[null,23,-19],"k":0,"mean_n":-87,"mean_d":5,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[23,\"2/25\"],[-19,\"1/25\"]],\"22/25\",\"-87/5\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0098","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[23, 2, 1]、D=26，權重為質量/D；X值=[-22, null, -20]。null是唯一未知槽，索引從0，k=1且該質量正；已知 E[X]=-239/13。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"24/1","explanation_zh":"未知整數值=24/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105098,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,null,-20],"k":1,"mean_n":-239,"mean_d":13,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[1,[[-22,\"23/26\"],null,[-20,\"1/26\"]],\"1/13\",\"-239/13\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0099","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[24, 2, 1]、D=27，權重為質量/D；X值=[-23, 25, null]。null是唯一未知槽，索引從0，k=2且該質量正；已知 E[X]=-523/27。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-21/1","explanation_zh":"未知整數值=-21/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105099,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,25,null],"k":2,"mean_n":-523,"mean_d":27,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[2,[[-23,\"8/9\"],[25,\"2/27\"],null],\"1/27\",\"-523/27\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r05-expectation-0100","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"給定均值的未知整數值","stem_zh":"有限歸一化模型；S={0,…,2}，F=𝒫(S)，質量=[25, 2, 1]、D=28，權重為質量/D；X值=[null, 26, -22]。null是唯一未知槽，索引從0，k=0且該質量正；已知 E[X]=-285/14。求該槽整數值，仍寫成p/1。","answer":{"kind":"expression","value":"-24/1","explanation_zh":"未知整數值=-24/1。以目標均值乘D扣已知質量值和，再除未知槽質量；正質量才可反解，本配方解為整數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["給定均值的未知整數值"]},"variant":{"family_id":"qf-density10-expectation-missing-value","seed":202610105100,"parameters":{"masses":[25,2,1],"D":28,"values":[null,26,-22],"k":0,"mean_n":-285,"mean_d":14,"operation":"density10_expectation_missing_value","domain":"finite_integer_random_variable","semantic_template":"qf-density10-expectation-missing-value/[0,[null,[26,\"1/14\"],[-22,\"1/28\"]],\"25/28\",\"-285/14\"]"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，不構成人工審定、普遍證明或形式 Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-08","contract_version":"ocme-question-factory-v0.1"}}