{"schema_version":"ocme-question-v0.1","id":"q-20261008-advance-20261010-r06-variance-0001","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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=[4, 99, 4]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1]","explanation_zh":"逐點中心平方項=[0/1, 0/1, 0/1]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106001,"parameters":{"masses":[1,0,2],"D":3,"values":[4,99,4],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[\"zero_variance_contributions\",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-r06-variance-0002","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 2]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[1/2, 1/2]","explanation_zh":"逐點中心平方項=[1/2, 1/2]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106002,"parameters":{"masses":[1,1],"D":2,"values":[0,2],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"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-r06-variance-0003","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 3]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[169/72, 121/108, 289/216]","explanation_zh":"逐點中心平方項=[169/72, 121/108, 289/216]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106003,"parameters":{"masses":[3,2,1],"D":6,"values":[-2,2,3],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"1/2\"],[\"-4\",\"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-r06-variance-0004","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 4]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[1156/343, 648/343, 1024/343]","explanation_zh":"逐點中心平方項=[1156/343, 648/343, 1024/343]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106004,"parameters":{"masses":[4,2,1],"D":7,"values":[-3,2,4],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"4/7\"],[\"-5\",\"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-r06-variance-0005","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 5]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[2205/512, 729/256, 2601/512]","explanation_zh":"逐點中心平方項=[2205/512, 729/256, 2601/512]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106005,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,2,5],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"5/8\"],[\"-6\",\"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-r06-variance-0006","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 6]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[1250/243, 2888/729, 5476/729]","explanation_zh":"逐點中心平方項=[1250/243, 2888/729, 5476/729]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106006,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,2,6],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"2/3\"],[\"-7\",\"2/9\"],[\"-11\",\"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-r06-variance-0007","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 7]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[5887/1000, 2601/500, 10201/1000]","explanation_zh":"逐點中心平方項=[5887/1000, 2601/500, 10201/1000]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106007,"parameters":{"masses":[7,2,1],"D":10,"values":[-6,2,7],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"7/10\"],[\"-8\",\"1/5\"],[\"-13\",\"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-r06-variance-0008","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 8]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[72/11, 72/11, 144/11]","explanation_zh":"逐點中心平方項=[72/11, 72/11, 144/11]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106008,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,2,8],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"8/11\"],[\"-9\",\"2/11\"],[\"-15\",\"1/11\"]]"}},"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-r06-variance-0009","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 9]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[1369/192, 6889/864, 27889/1728]","explanation_zh":"逐點中心平方項=[1369/192, 6889/864, 27889/1728]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106009,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,2,9],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"3/4\"],[\"-10\",\"1/6\"],[\"-17\",\"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-r06-variance-0010","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 10]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[16810/2197, 20808/2197, 42436/2197]","explanation_zh":"逐點中心平方項=[16810/2197, 20808/2197, 42436/2197]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106010,"parameters":{"masses":[10,2,1],"D":13,"values":[-9,2,10],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"10/13\"],[\"-11\",\"2/13\"],[\"-19\",\"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-r06-variance-0011","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 11]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[22275/2744, 15129/1372, 62001/2744]","explanation_zh":"逐點中心平方項=[22275/2744, 15129/1372, 62001/2744]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106011,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,2,11],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"11/14\"],[\"-12\",\"1/7\"],[\"-21\",\"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-r06-variance-0012","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 12]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[9604/1125, 42632/3375, 87616/3375]","explanation_zh":"逐點中心平方項=[9604/1125, 42632/3375, 87616/3375]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106012,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,2,12],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"4/5\"],[\"-13\",\"2/15\"],[\"-23\",\"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-r06-variance-0013","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 13]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[36517/4096, 29241/2048, 120409/4096]","explanation_zh":"逐點中心平方項=[36517/4096, 29241/2048, 120409/4096]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106013,"parameters":{"masses":[13,2,1],"D":16,"values":[-12,2,13],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"13/16\"],[\"-14\",\"1/8\"],[\"-25\",\"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-r06-variance-0014","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 14]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[45486/4913, 78408/4913, 161604/4913]","explanation_zh":"逐點中心平方項=[45486/4913, 78408/4913, 161604/4913]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106014,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,2,14],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"14/17\"],[\"-15\",\"2/17\"],[\"-27\",\"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-r06-variance-0015","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 15]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[18605/1944, 51529/2916, 212521/5832]","explanation_zh":"逐點中心平方項=[18605/1944, 51529/2916, 212521/5832]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106015,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,2,15],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"5/6\"],[\"-16\",\"1/9\"],[\"-29\",\"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-r06-variance-0016","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 16]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[67600/6859, 133128/6859, 274576/6859]","explanation_zh":"逐點中心平方項=[67600/6859, 133128/6859, 274576/6859]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106016,"parameters":{"masses":[16,2,1],"D":19,"values":[-15,2,16],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"16/19\"],[\"-17\",\"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-r06-variance-0017","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 17]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[80937/8000, 84681/4000, 349281/8000]","explanation_zh":"逐點中心平方項=[80937/8000, 84681/4000, 349281/8000]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106017,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,2,17],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"17/20\"],[\"-18\",\"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-r06-variance-0018","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 18]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[10658/1029, 212552/9261, 438244/9261]","explanation_zh":"逐點中心平方項=[10658/1029, 212552/9261, 438244/9261]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106018,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,2,18],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"6/7\"],[\"-19\",\"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-r06-variance-0019","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 19]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[931/88, 99/4, 4489/88]","explanation_zh":"逐點中心平方項=[931/88, 99/4, 4489/88]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106019,"parameters":{"masses":[19,2,1],"D":22,"values":[-18,2,19],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"19/22\"],[\"-20\",\"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-r06-variance-0020","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 20]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[131220/12167, 323208/12167, 665856/12167]","explanation_zh":"逐點中心平方項=[131220/12167, 323208/12167, 665856/12167]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106020,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,2,20],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"20/23\"],[\"-21\",\"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-r06-variance-0021","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 21]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[50575/4608, 196249/6912, 808201/13824]","explanation_zh":"逐點中心平方項=[50575/4608, 196249/6912, 808201/13824]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106021,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,2,21],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"7/8\"],[\"-22\",\"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-r06-variance-0022","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 22]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[174262/15625, 472392/15625, 972196/15625]","explanation_zh":"逐點中心平方項=[174262/15625, 472392/15625, 972196/15625]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106022,"parameters":{"masses":[22,2,1],"D":25,"values":[-21,2,22],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"22/25\"],[\"-23\",\"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-r06-variance-0023","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 23]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[198927/17576, 281961/8788, 1159929/17576]","explanation_zh":"逐點中心平方項=[198927/17576, 281961/8788, 1159929/17576]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106023,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,2,23],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"23/26\"],[\"-24\",\"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-r06-variance-0024","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 24]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[75272/6561, 668168/19683, 1373584/19683]","explanation_zh":"逐點中心平方項=[75272/6561, 668168/19683, 1373584/19683]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106024,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,2,24],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"8/9\"],[\"-25\",\"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-r06-variance-0025","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"逐點的中心平方貢獻","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, 25]。先取μ=E[X]，按索引0起列 w_i(X_i−μ)²，不省零項。","answer":{"kind":"expression","value":"[255025/21952, 393129/10976, 1615441/21952]","explanation_zh":"逐點中心平方項=[255025/21952, 393129/10976, 1615441/21952]。先求精確均值再平方偏差與加權；零權重異值保留零項，這是概率量，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["逐點的中心平方貢獻"]},"variant":{"family_id":"qf-density10-variance-contributions","seed":202610106025,"parameters":{"masses":[25,2,1],"D":28,"values":[-24,2,25],"operation":"density10_variance_contributions","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-contributions/[[\"0\",\"25/28\"],[\"-26\",\"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-r06-variance-0026","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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=[4, 99, 4]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(16/1, 16/1)","explanation_zh":"二階矩與均值平方=(16/1, 16/1)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106026,"parameters":{"masses":[1,0,2],"D":3,"values":[4,99,4],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-4\",\"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-r06-variance-0027","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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=[-1, 1]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(1/1, 0/1)","explanation_zh":"二階矩與均值平方=(1/1, 0/1)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106027,"parameters":{"masses":[1,1],"D":2,"values":[-1,1],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-1\",\"1/2\"],[\"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-r06-variance-0028","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(29/6, 1/36)","explanation_zh":"二階矩與均值平方=(29/6, 1/36)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106028,"parameters":{"masses":[3,2,1],"D":6,"values":[-2,2,3],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/2\"],[\"2\",\"1/3\"],[\"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-r06-variance-0029","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(60/7, 16/49)","explanation_zh":"二階矩與均值平方=(60/7, 16/49)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106029,"parameters":{"masses":[4,2,1],"D":7,"values":[-3,2,4],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/7\"],[\"-4\",\"1/7\"],[\"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-r06-variance-0030","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(113/8, 121/64)","explanation_zh":"二階矩與均值平方=(113/8, 121/64)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106030,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,2,5],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/4\"],[\"-5\",\"1/8\"],[\"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-r06-variance-0031","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(194/9, 400/81)","explanation_zh":"二階矩與均值平方=(194/9, 400/81)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106031,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,2,6],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/9\"],[\"-6\",\"1/9\"],[\"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-r06-variance-0032","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(309/10, 961/100)","explanation_zh":"二階矩與均值平方=(309/10, 961/100)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106032,"parameters":{"masses":[7,2,1],"D":10,"values":[-6,2,7],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/5\"],[\"-7\",\"1/10\"],[\"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-r06-variance-0033","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(464/11, 16/1)","explanation_zh":"二階矩與均值平方=(464/11, 16/1)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106033,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,2,8],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/11\"],[\"-8\",\"1/11\"],[\"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-r06-variance-0034","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(665/12, 3481/144)","explanation_zh":"二階矩與均值平方=(665/12, 3481/144)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106034,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,2,9],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/6\"],[\"-9\",\"1/12\"],[\"8\",\"3/4\"]]"}},"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-r06-variance-0035","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(918/13, 5776/169)","explanation_zh":"二階矩與均值平方=(918/13, 5776/169)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106035,"parameters":{"masses":[10,2,1],"D":13,"values":[-9,2,10],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-10\",\"1/13\"],[\"-2\",\"2/13\"],[\"9\",\"10/13\"]]"}},"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-r06-variance-0036","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(1229/14, 9025/196)","explanation_zh":"二階矩與均值平方=(1229/14, 9025/196)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106036,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,2,11],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-10\",\"11/14\"],[\"11\",\"1/14\"],[\"2\",\"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-r06-variance-0037","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(1604/15, 13456/225)","explanation_zh":"二階矩與均值平方=(1604/15, 13456/225)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106037,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,2,12],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-11\",\"4/5\"],[\"12\",\"1/15\"],[\"2\",\"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-r06-variance-0038","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(2049/16, 19321/256)","explanation_zh":"二階矩與均值平方=(2049/16, 19321/256)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106038,"parameters":{"masses":[13,2,1],"D":16,"values":[-12,2,13],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-12\",\"13/16\"],[\"13\",\"1/16\"],[\"2\",\"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-r06-variance-0039","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(2570/17, 26896/289)","explanation_zh":"二階矩與均值平方=(2570/17, 26896/289)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106039,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,2,14],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-13\",\"14/17\"],[\"14\",\"1/17\"],[\"2\",\"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-r06-variance-0040","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(3173/18, 36481/324)","explanation_zh":"二階矩與均值平方=(3173/18, 36481/324)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106040,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,2,15],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-14\",\"5/6\"],[\"15\",\"1/18\"],[\"2\",\"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-r06-variance-0041","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(3864/19, 48400/361)","explanation_zh":"二階矩與均值平方=(3864/19, 48400/361)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106041,"parameters":{"masses":[16,2,1],"D":19,"values":[-15,2,16],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-15\",\"16/19\"],[\"16\",\"1/19\"],[\"2\",\"2/19\"]]"}},"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-r06-variance-0042","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(4649/20, 63001/400)","explanation_zh":"二階矩與均值平方=(4649/20, 63001/400)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106042,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,2,17],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-16\",\"17/20\"],[\"17\",\"1/20\"],[\"2\",\"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-r06-variance-0043","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(5534/21, 80656/441)","explanation_zh":"二階矩與均值平方=(5534/21, 80656/441)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106043,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,2,18],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-17\",\"6/7\"],[\"18\",\"1/21\"],[\"2\",\"2/21\"]]"}},"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-r06-variance-0044","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(6525/22, 841/4)","explanation_zh":"二階矩與均值平方=(6525/22, 841/4)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106044,"parameters":{"masses":[19,2,1],"D":22,"values":[-18,2,19],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-18\",\"19/22\"],[\"19\",\"1/22\"],[\"2\",\"1/11\"]]"}},"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-r06-variance-0045","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(7628/23, 126736/529)","explanation_zh":"二階矩與均值平方=(7628/23, 126736/529)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106045,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,2,20],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-19\",\"20/23\"],[\"2\",\"2/23\"],[\"20\",\"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-r06-variance-0046","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(8849/24, 156025/576)","explanation_zh":"二階矩與均值平方=(8849/24, 156025/576)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106046,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,2,21],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/12\"],[\"-21\",\"1/24\"],[\"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-r06-variance-0047","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(10194/25, 190096/625)","explanation_zh":"二階矩與均值平方=(10194/25, 190096/625)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106047,"parameters":{"masses":[22,2,1],"D":25,"values":[-21,2,22],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/25\"],[\"-22\",\"1/25\"],[\"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-r06-variance-0048","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(11669/26, 229441/676)","explanation_zh":"二階矩與均值平方=(11669/26, 229441/676)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106048,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,2,23],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/13\"],[\"-23\",\"1/26\"],[\"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-r06-variance-0049","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(13280/27, 274576/729)","explanation_zh":"二階矩與均值平方=(13280/27, 274576/729)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106049,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,2,24],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/27\"],[\"-24\",\"1/27\"],[\"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-r06-variance-0050","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"二階矩與均值平方的有序對","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]。依序求 (E[X²], (E[X])²)，不要混同兩式。","answer":{"kind":"expression","value":"(15033/28, 326041/784)","explanation_zh":"二階矩與均值平方=(15033/28, 326041/784)。第一式平方值後加權，第二式加權均值後平方；兩式之差才是變異數，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["二階矩與均值平方的有序對"]},"variant":{"family_id":"qf-density10-variance-second-moment-pair","seed":202610106050,"parameters":{"masses":[25,2,1],"D":28,"values":[-24,2,25],"operation":"density10_variance_second_moment_pair","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/14\"],[\"-25\",\"1/28\"],[\"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-r06-variance-0051","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"0/1","explanation_zh":"共模型差變異數=0/1。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106051,"parameters":{"masses":[1,0,2],"D":3,"values":[4,99,4],"other":[4,-50,4],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-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-r06-variance-0052","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"4/1","explanation_zh":"共模型差變異數=4/1。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106052,"parameters":{"masses":[1,1],"D":2,"values":[0,2],"other":[2,0],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-4\",\"1/2\"],[\"0\",\"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-r06-variance-0053","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"104/9","explanation_zh":"共模型差變異數=104/9。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106053,"parameters":{"masses":[3,2,1],"D":6,"values":[-2,2,3],"other":[1,-1,-2],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-6\",\"1/3\"],[\"-8\",\"1/6\"],[\"0\",\"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-r06-variance-0054","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"908/49","explanation_zh":"共模型差變異數=908/49。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106054,"parameters":{"masses":[4,2,1],"D":7,"values":[-3,2,4],"other":[1,-1,-3],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-11\",\"1/7\"],[\"-7\",\"2/7\"],[\"0\",\"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-r06-variance-0055","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"423/16","explanation_zh":"共模型差變異數=423/16。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106055,"parameters":{"masses":[5,2,1],"D":8,"values":[-4,2,5],"other":[1,-1,-4],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-14\",\"1/8\"],[\"-8\",\"1/4\"],[\"0\",\"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-r06-variance-0056","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"2834/81","explanation_zh":"共模型差變異數=2834/81。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106056,"parameters":{"masses":[6,2,1],"D":9,"values":[-5,2,6],"other":[1,-1,-5],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-17\",\"1/9\"],[\"-9\",\"2/9\"],[\"0\",\"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-r06-variance-0057","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"44/1","explanation_zh":"共模型差變異數=44/1。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106057,"parameters":{"masses":[7,2,1],"D":10,"values":[-6,2,7],"other":[1,-1,-6],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-10\",\"1/5\"],[\"-20\",\"1/10\"],[\"0\",\"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-r06-variance-0058","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"6456/121","explanation_zh":"共模型差變異數=6456/121。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106058,"parameters":{"masses":[8,2,1],"D":11,"values":[-7,2,8],"other":[1,-1,-7],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-11\",\"2/11\"],[\"-23\",\"1/11\"],[\"0\",\"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-r06-variance-0059","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"2267/36","explanation_zh":"共模型差變異數=2267/36。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106059,"parameters":{"masses":[9,2,1],"D":12,"values":[-8,2,9],"other":[1,-1,-8],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-12\",\"1/6\"],[\"-26\",\"1/12\"],[\"0\",\"3/4\"]]"}},"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-r06-variance-0060","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"12302/169","explanation_zh":"共模型差變異數=12302/169。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106060,"parameters":{"masses":[10,2,1],"D":13,"values":[-9,2,10],"other":[1,-1,-9],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-13\",\"2/13\"],[\"-29\",\"1/13\"],[\"0\",\"10/13\"]]"}},"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-r06-variance-0061","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"4056/49","explanation_zh":"共模型差變異數=4056/49。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106061,"parameters":{"masses":[11,2,1],"D":14,"values":[-10,2,11],"other":[1,-1,-10],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-14\",\"1/7\"],[\"-32\",\"1/14\"],[\"0\",\"11/14\"]]"}},"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-r06-variance-0062","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"836/9","explanation_zh":"共模型差變異數=836/9。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106062,"parameters":{"masses":[12,2,1],"D":15,"values":[-11,2,12],"other":[1,-1,-11],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-15\",\"2/15\"],[\"-35\",\"1/15\"],[\"0\",\"4/5\"]]"}},"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-r06-variance-0063","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"6599/64","explanation_zh":"共模型差變異數=6599/64。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106063,"parameters":{"masses":[13,2,1],"D":16,"values":[-12,2,13],"other":[1,-1,-12],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-16\",\"1/8\"],[\"-38\",\"1/16\"],[\"0\",\"13/16\"]]"}},"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-r06-variance-0064","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"32778/289","explanation_zh":"共模型差變異數=32778/289。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106064,"parameters":{"masses":[14,2,1],"D":17,"values":[-13,2,14],"other":[1,-1,-13],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-17\",\"2/17\"],[\"-41\",\"1/17\"],[\"0\",\"14/17\"]]"}},"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-r06-variance-0065","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"10028/81","explanation_zh":"共模型差變異數=10028/81。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106065,"parameters":{"masses":[15,2,1],"D":18,"values":[-14,2,15],"other":[1,-1,-14],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-18\",\"1/9\"],[\"-44\",\"1/18\"],[\"0\",\"5/6\"]]"}},"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-r06-variance-0066","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"48464/361","explanation_zh":"共模型差變異數=48464/361。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106066,"parameters":{"masses":[16,2,1],"D":19,"values":[-15,2,16],"other":[1,-1,-15],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-19\",\"2/19\"],[\"-47\",\"1/19\"],[\"0\",\"16/19\"]]"}},"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-r06-variance-0067","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"579/4","explanation_zh":"共模型差變異數=579/4。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106067,"parameters":{"masses":[17,2,1],"D":20,"values":[-16,2,17],"other":[1,-1,-16],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-20\",\"1/10\"],[\"-50\",\"1/20\"],[\"0\",\"17/20\"]]"}},"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-r06-variance-0068","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"68486/441","explanation_zh":"共模型差變異數=68486/441。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106068,"parameters":{"masses":[18,2,1],"D":21,"values":[-17,2,18],"other":[1,-1,-17],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-21\",\"2/21\"],[\"-53\",\"1/21\"],[\"0\",\"6/7\"]]"}},"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-r06-variance-0069","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"20072/121","explanation_zh":"共模型差變異數=20072/121。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106069,"parameters":{"masses":[19,2,1],"D":22,"values":[-18,2,19],"other":[1,-1,-18],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-22\",\"1/11\"],[\"-56\",\"1/22\"],[\"0\",\"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-r06-variance-0070","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"93372/529","explanation_zh":"共模型差變異數=93372/529。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106070,"parameters":{"masses":[20,2,1],"D":23,"values":[-19,2,20],"other":[1,-1,-19],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-23\",\"2/23\"],[\"-59\",\"1/23\"],[\"0\",\"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-r06-variance-0071","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"26951/144","explanation_zh":"共模型差變異數=26951/144。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106071,"parameters":{"masses":[21,2,1],"D":24,"values":[-20,2,21],"other":[1,-1,-20],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-24\",\"1/12\"],[\"-62\",\"1/24\"],[\"0\",\"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-r06-variance-0072","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"4946/25","explanation_zh":"共模型差變異數=4946/25。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106072,"parameters":{"masses":[22,2,1],"D":25,"values":[-21,2,22],"other":[1,-1,-21],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-25\",\"2/25\"],[\"-65\",\"1/25\"],[\"0\",\"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-r06-variance-0073","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"35244/169","explanation_zh":"共模型差變異數=35244/169。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106073,"parameters":{"masses":[23,2,1],"D":26,"values":[-22,2,23],"other":[1,-1,-22],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-26\",\"1/13\"],[\"-68\",\"1/26\"],[\"0\",\"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-r06-variance-0074","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"159848/729","explanation_zh":"共模型差變異數=159848/729。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106074,"parameters":{"masses":[24,2,1],"D":27,"values":[-23,2,24],"other":[1,-1,-23],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-27\",\"2/27\"],[\"-71\",\"1/27\"],[\"0\",\"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-r06-variance-0075","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"共模型差變數的概率變異數","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 Var(X−Y)，不要拆作Var(X)+Var(Y)。","answer":{"kind":"expression","value":"45083/196","explanation_zh":"共模型差變異數=45083/196。先逐點作差，再用各點對的加權差平方和除D²；相依值可使差在正支持恆定，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共模型差變數的概率變異數"]},"variant":{"family_id":"qf-density10-variance-difference","seed":202610106075,"parameters":{"masses":[25,2,1],"D":28,"values":[-24,2,25],"other":[1,-1,-24],"operation":"density10_variance_difference","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-difference/[[\"-28\",\"1/14\"],[\"-74\",\"1/28\"],[\"0\",\"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-r06-variance-0076","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 2, 0, 1, 0, 0, 0, 0]、D=3，權重為質量/D；X值=[99, 4, -99, 4, 9, 8, 7, 6]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"無","explanation_zh":"正支持首個失配索引=無。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106076,"parameters":{"masses":[0,2,0,1,0,0,0,0],"D":3,"values":[99,4,-99,4,9,8,7,6],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[\"constant_positive_support\"]"}},"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-r06-variance-0077","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[2, 2, 0, 0, 0, 0, 0, 0]、D=4，權重為質量/D；X值=[0, 2, 102, 103, 104, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"1","explanation_zh":"正支持首個失配索引=1。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106077,"parameters":{"masses":[2,2,0,0,0,0,0,0],"D":4,"values":[0,2,102,103,104,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0078","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 3, 2, 0, 0, 0, 0, 0]、D=5，權重為質量/D；X值=[100, 0, 3, 103, 104, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"2","explanation_zh":"正支持首個失配索引=2。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106078,"parameters":{"masses":[0,3,2,0,0,0,0,0],"D":5,"values":[100,0,3,103,104,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0079","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 4, 2, 0, 0, 0, 0]、D=6，權重為質量/D；X值=[100, 101, 0, 4, 104, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"3","explanation_zh":"正支持首個失配索引=3。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106079,"parameters":{"masses":[0,0,4,2,0,0,0,0],"D":6,"values":[100,101,0,4,104,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0080","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 5, 2, 0, 0, 0]、D=7，權重為質量/D；X值=[100, 101, 102, 0, 5, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"4","explanation_zh":"正支持首個失配索引=4。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106080,"parameters":{"masses":[0,0,0,5,2,0,0,0],"D":7,"values":[100,101,102,0,5,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[3,4]"}},"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-r06-variance-0081","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 0, 6, 2, 0, 0]、D=8，權重為質量/D；X值=[100, 101, 102, 103, 0, 6, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"5","explanation_zh":"正支持首個失配索引=5。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106081,"parameters":{"masses":[0,0,0,0,6,2,0,0],"D":8,"values":[100,101,102,103,0,6,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[4,5]"}},"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-r06-variance-0082","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 0, 0, 7, 2, 0]、D=9，權重為質量/D；X值=[100, 101, 102, 103, 104, 0, 7, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"6","explanation_zh":"正支持首個失配索引=6。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106082,"parameters":{"masses":[0,0,0,0,0,7,2,0],"D":9,"values":[100,101,102,103,104,0,7,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[5,6]"}},"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-r06-variance-0083","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 0, 0, 0, 8, 2]、D=10，權重為質量/D；X值=[100, 101, 102, 103, 104, 105, 0, 8]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"7","explanation_zh":"正支持首個失配索引=7。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106083,"parameters":{"masses":[0,0,0,0,0,0,8,2],"D":10,"values":[100,101,102,103,104,105,0,8],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[6,7]"}},"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-r06-variance-0084","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[9, 0, 2, 0, 0, 0, 0, 0]、D=11，權重為質量/D；X值=[0, 101, 9, 103, 104, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"2","explanation_zh":"正支持首個失配索引=2。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106084,"parameters":{"masses":[9,0,2,0,0,0,0,0],"D":11,"values":[0,101,9,103,104,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[0,2]"}},"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-r06-variance-0085","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 10, 0, 2, 0, 0, 0, 0]、D=12，權重為質量/D；X值=[100, 0, 102, 10, 104, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"3","explanation_zh":"正支持首個失配索引=3。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106085,"parameters":{"masses":[0,10,0,2,0,0,0,0],"D":12,"values":[100,0,102,10,104,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0086","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 11, 0, 2, 0, 0, 0]、D=13，權重為質量/D；X值=[100, 101, 0, 103, 11, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"4","explanation_zh":"正支持首個失配索引=4。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106086,"parameters":{"masses":[0,0,11,0,2,0,0,0],"D":13,"values":[100,101,0,103,11,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[2,4]"}},"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-r06-variance-0087","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 12, 0, 2, 0, 0]、D=14，權重為質量/D；X值=[100, 101, 102, 0, 104, 12, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"5","explanation_zh":"正支持首個失配索引=5。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106087,"parameters":{"masses":[0,0,0,12,0,2,0,0],"D":14,"values":[100,101,102,0,104,12,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[3,5]"}},"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-r06-variance-0088","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 0, 13, 0, 2, 0]、D=15，權重為質量/D；X值=[100, 101, 102, 103, 0, 105, 13, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"6","explanation_zh":"正支持首個失配索引=6。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106088,"parameters":{"masses":[0,0,0,0,13,0,2,0],"D":15,"values":[100,101,102,103,0,105,13,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[4,6]"}},"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-r06-variance-0089","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 0, 0, 14, 0, 2]、D=16，權重為質量/D；X值=[100, 101, 102, 103, 104, 0, 106, 14]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"7","explanation_zh":"正支持首個失配索引=7。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106089,"parameters":{"masses":[0,0,0,0,0,14,0,2],"D":16,"values":[100,101,102,103,104,0,106,14],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[5,7]"}},"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-r06-variance-0090","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[15, 0, 0, 2, 0, 0, 0, 0]、D=17，權重為質量/D；X值=[0, 101, 102, 15, 104, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"3","explanation_zh":"正支持首個失配索引=3。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106090,"parameters":{"masses":[15,0,0,2,0,0,0,0],"D":17,"values":[0,101,102,15,104,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[0,3]"}},"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-r06-variance-0091","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 16, 0, 0, 2, 0, 0, 0]、D=18，權重為質量/D；X值=[100, 0, 102, 103, 16, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"4","explanation_zh":"正支持首個失配索引=4。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106091,"parameters":{"masses":[0,16,0,0,2,0,0,0],"D":18,"values":[100,0,102,103,16,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[1,4]"}},"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-r06-variance-0092","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 17, 0, 0, 2, 0, 0]、D=19，權重為質量/D；X值=[100, 101, 0, 103, 104, 17, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"5","explanation_zh":"正支持首個失配索引=5。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106092,"parameters":{"masses":[0,0,17,0,0,2,0,0],"D":19,"values":[100,101,0,103,104,17,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[2,5]"}},"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-r06-variance-0093","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 18, 0, 0, 2, 0]、D=20，權重為質量/D；X值=[100, 101, 102, 0, 104, 105, 18, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"6","explanation_zh":"正支持首個失配索引=6。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106093,"parameters":{"masses":[0,0,0,18,0,0,2,0],"D":20,"values":[100,101,102,0,104,105,18,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[3,6]"}},"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-r06-variance-0094","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 0, 19, 0, 0, 2]、D=21，權重為質量/D；X值=[100, 101, 102, 103, 0, 105, 106, 19]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"7","explanation_zh":"正支持首個失配索引=7。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106094,"parameters":{"masses":[0,0,0,0,19,0,0,2],"D":21,"values":[100,101,102,103,0,105,106,19],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0095","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[20, 0, 0, 0, 2, 0, 0, 0]、D=22，權重為質量/D；X值=[0, 101, 102, 103, 20, 105, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"4","explanation_zh":"正支持首個失配索引=4。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106095,"parameters":{"masses":[20,0,0,0,2,0,0,0],"D":22,"values":[0,101,102,103,20,105,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[0,4]"}},"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-r06-variance-0096","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 21, 0, 0, 0, 2, 0, 0]、D=23，權重為質量/D；X值=[100, 0, 102, 103, 104, 21, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"5","explanation_zh":"正支持首個失配索引=5。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106096,"parameters":{"masses":[0,21,0,0,0,2,0,0],"D":23,"values":[100,0,102,103,104,21,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0097","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 22, 0, 0, 0, 2, 0]、D=24，權重為質量/D；X值=[100, 101, 0, 103, 104, 105, 22, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"6","explanation_zh":"正支持首個失配索引=6。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106097,"parameters":{"masses":[0,0,22,0,0,0,2,0],"D":24,"values":[100,101,0,103,104,105,22,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[2,6]"}},"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-r06-variance-0098","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 0, 0, 23, 0, 0, 0, 2]、D=25，權重為質量/D；X值=[100, 101, 102, 0, 104, 105, 106, 23]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"7","explanation_zh":"正支持首個失配索引=7。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106098,"parameters":{"masses":[0,0,0,23,0,0,0,2],"D":25,"values":[100,101,102,0,104,105,106,23],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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-r06-variance-0099","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[24, 0, 0, 0, 0, 2, 0, 0]、D=26，權重為質量/D；X值=[0, 101, 102, 103, 104, 24, 106, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"5","explanation_zh":"正支持首個失配索引=5。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106099,"parameters":{"masses":[24,0,0,0,0,2,0,0],"D":26,"values":[0,101,102,103,104,24,106,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[0,5]"}},"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-r06-variance-0100","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正支持相對起點的首個失配","stem_zh":"有限歸一化模型；S={0,…,7}，F=𝒫(S)，質量=[0, 25, 0, 0, 0, 0, 2, 0]、D=27，權重為質量/D；X值=[100, 0, 102, 103, 104, 105, 25, 107]。索引從0；以第一個正質量點的X值為基準，列第一個正質量且X值不同的索引，若無則答無。","answer":{"kind":"short_text","value":"6","explanation_zh":"正支持首個失配索引=6。只比較正質量支持，零權重異值不列失配；全部正支持相同給無，非樣本估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正支持相對起點的首個失配"]},"variant":{"family_id":"qf-density10-variance-first-support-mismatch","seed":202610106100,"parameters":{"masses":[0,25,0,0,0,0,2,0],"D":27,"values":[100,0,102,103,104,105,25,107],"operation":"density10_variance_first_support_mismatch","domain":"finite_integer_random_variable","semantic_template":"qf-density10-variance-first-support-mismatch/[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"}}