{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0001","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 5), (1, 5), (2, 999)}，各結果依列序 [0, 1, 2] 的非負機率權重為 [1/3, 2/3, 0/3]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"變異數=0/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057001,"parameters":{"S":[0,1,2],"F":"powerset(S)","values":[5,5,999],"weights":[1,2,0],"denominator":3,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/1\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0002","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, -4), (3, 1000)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/8, 3/8, 4/8, 0/8]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"9/4","explanation_zh":"變異數=9/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057002,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"3\",\"1/2\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0003","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, -4), (3, 1001)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 4/10, 5/10, 0/10]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"4/1","explanation_zh":"變異數=4/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057003,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"4\",\"1/2\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0004","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, -4), (3, 1002)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 5/12, 6/12, 0/12]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"25/4","explanation_zh":"變異數=25/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057004,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"5\",\"1/2\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0005","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 0), (3, 1003)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 6/10, 3/10, 0/10]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"189/25","explanation_zh":"變異數=189/25。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057005,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"7/10\"],[\"6\",\"3/10\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0006","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"98/9","explanation_zh":"變異數=98/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057006,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"2/3\"],[\"7\",\"1/3\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0007","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 4), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 8/14, 5/14, 0/14]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"720/49","explanation_zh":"變異數=720/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057007,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"9/14\"],[\"8\",\"5/14\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0008","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, -4), (3, 1006)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 9/16, 6/16, 0/16]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1215/64","explanation_zh":"變異數=1215/64。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057008,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/8\"],[\"9\",\"3/8\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0009","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 8), (3, 1007)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 10/14, 3/14, 0/14]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"825/49","explanation_zh":"變異數=825/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057009,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"11/14\"],[\"10\",\"3/14\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0010","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, -4), (3, 1008)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 11/16, 4/16, 0/16]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"363/16","explanation_zh":"變異數=363/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057010,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"3/4\"],[\"11\",\"1/4\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0011","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0), (1, 0), (2, 12), (3, 1009)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 12/18, 5/18, 0/18]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"260/9","explanation_zh":"變異數=260/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057011,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/18\"],[\"12\",\"5/18\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0012","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, -4), (3, 1010)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 13/20, 6/20, 0/20]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"3549/100","explanation_zh":"變異數=3549/100。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057012,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"7/10\"],[\"13\",\"3/10\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0013","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 2), (1, 2), (2, 16), (3, 1011)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 14/18, 3/18, 0/18]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"245/9","explanation_zh":"變異數=245/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057013,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"14\",\"1/6\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0014","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, -4), (3, 1012)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 15/20, 4/20, 0/20]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"36/1","explanation_zh":"變異數=36/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057014,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"4/5\"],[\"15\",\"1/5\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0015","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 4), (1, 4), (2, 20), (3, 1013)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 16/22, 5/22, 0/22]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"5440/121","explanation_zh":"變異數=5440/121。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057015,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"17/22\"],[\"16\",\"5/22\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0016","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, -4), (3, 1014)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 17/24, 6/24, 0/24]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"867/16","explanation_zh":"變異數=867/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057016,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"3/4\"],[\"17\",\"1/4\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0017","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 6), (1, 6), (2, 24), (3, 1015)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 18/22, 3/22, 0/22]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"4617/121","explanation_zh":"變異數=4617/121。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057017,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"19/22\"],[\"18\",\"3/22\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0018","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, -4), (3, 1016)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 19/24, 4/24, 0/24]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1805/36","explanation_zh":"變異數=1805/36。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057018,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"19\",\"1/6\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0019","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 8), (1, 8), (2, 28), (3, 1017)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 20/26, 5/26, 0/26]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"10500/169","explanation_zh":"變異數=10500/169。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057019,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"21/26\"],[\"20\",\"5/26\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0020","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -25), (1, -25), (2, -4), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 21/28, 6/28, 0/28]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"297/4","explanation_zh":"變異數=297/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057020,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"11/14\"],[\"21\",\"3/14\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0021","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 10), (1, 10), (2, 32), (3, 1019)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 22/26, 3/26, 0/26]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"8349/169","explanation_zh":"變異數=8349/169。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057021,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"23/26\"],[\"22\",\"3/26\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0022","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -27), (1, -27), (2, -4), (3, 1020)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 23/28, 4/28, 0/28]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"3174/49","explanation_zh":"變異數=3174/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057022,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"6/7\"],[\"23\",\"1/7\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0023","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 12), (1, 12), (2, 36), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 24/30, 5/30, 0/30]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"80/1","explanation_zh":"變異數=80/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057023,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"24\",\"1/6\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0024","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -29), (1, -29), (2, -4), (3, 1022)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/32, 25/32, 6/32, 0/32]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"24375/256","explanation_zh":"變異數=24375/256。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057024,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/16\"],[\"25\",\"3/16\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0025","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"精確有限機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 14), (1, 14), (2, 40), (3, 1023)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 26/30, 3/30, 0/30]，權重總和恰為一。令 Z=X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1521/25","explanation_zh":"變異數=1521/25。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["精確有限機率變異數"]},"variant":{"family_id":"qf-variance-weighted","seed":202610057025,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"operation":"variance","semantic_template":"qf-variance-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"9/10\"],[\"26\",\"1/10\"]],\"operation\":\"variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0026","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 5), (1, 5), (2, 999)}，各結果依列序 [0, 1, 2] 的非負機率權重為 [1/3, 2/3, 0/3]，權重總和恰為一。令 Z=X+(-12)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"變異數=0/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057026,"parameters":{"S":[0,1,2],"F":"powerset(S)","values":[5,5,999],"weights":[1,2,0],"denominator":3,"c":-12,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/1\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0027","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, -4), (3, 1000)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/8, 3/8, 4/8, 0/8]，權重總和恰為一。令 Z=X+(-11)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"9/4","explanation_zh":"變異數=9/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057027,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"c":-11,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"3\",\"1/2\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0028","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, -4), (3, 1001)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 4/10, 5/10, 0/10]，權重總和恰為一。令 Z=X+(-10)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"4/1","explanation_zh":"變異數=4/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057028,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"c":-10,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"4\",\"1/2\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0029","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, -4), (3, 1002)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 5/12, 6/12, 0/12]，權重總和恰為一。令 Z=X+(-9)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"25/4","explanation_zh":"變異數=25/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057029,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"c":-9,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"5\",\"1/2\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0030","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 0), (3, 1003)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 6/10, 3/10, 0/10]，權重總和恰為一。令 Z=X+(-8)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"189/25","explanation_zh":"變異數=189/25。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057030,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"c":-8,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"7/10\"],[\"6\",\"3/10\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0031","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=X+(-7)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"98/9","explanation_zh":"變異數=98/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057031,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"c":-7,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"2/3\"],[\"7\",\"1/3\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0032","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 4), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 8/14, 5/14, 0/14]，權重總和恰為一。令 Z=X+(-6)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"720/49","explanation_zh":"變異數=720/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057032,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"c":-6,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"9/14\"],[\"8\",\"5/14\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0033","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, -4), (3, 1006)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 9/16, 6/16, 0/16]，權重總和恰為一。令 Z=X+(-5)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1215/64","explanation_zh":"變異數=1215/64。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057033,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"c":-5,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/8\"],[\"9\",\"3/8\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0034","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 8), (3, 1007)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 10/14, 3/14, 0/14]，權重總和恰為一。令 Z=X+(-4)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"825/49","explanation_zh":"變異數=825/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057034,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"c":-4,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"11/14\"],[\"10\",\"3/14\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0035","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, -4), (3, 1008)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 11/16, 4/16, 0/16]，權重總和恰為一。令 Z=X+(-3)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"363/16","explanation_zh":"變異數=363/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057035,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"c":-3,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"3/4\"],[\"11\",\"1/4\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0036","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0), (1, 0), (2, 12), (3, 1009)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 12/18, 5/18, 0/18]，權重總和恰為一。令 Z=X+(-2)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"260/9","explanation_zh":"變異數=260/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057036,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"c":-2,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/18\"],[\"12\",\"5/18\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0037","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, -4), (3, 1010)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 13/20, 6/20, 0/20]，權重總和恰為一。令 Z=X+(-1)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"3549/100","explanation_zh":"變異數=3549/100。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057037,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"c":-1,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"7/10\"],[\"13\",\"3/10\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0038","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 2), (1, 2), (2, 16), (3, 1011)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 14/18, 3/18, 0/18]，權重總和恰為一。令 Z=X+(0)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"245/9","explanation_zh":"變異數=245/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057038,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"c":0,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"14\",\"1/6\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0039","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, -4), (3, 1012)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 15/20, 4/20, 0/20]，權重總和恰為一。令 Z=X+(1)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"36/1","explanation_zh":"變異數=36/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057039,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"c":1,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"4/5\"],[\"15\",\"1/5\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0040","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 4), (1, 4), (2, 20), (3, 1013)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 16/22, 5/22, 0/22]，權重總和恰為一。令 Z=X+(2)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"5440/121","explanation_zh":"變異數=5440/121。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057040,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"c":2,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"17/22\"],[\"16\",\"5/22\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0041","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, -4), (3, 1014)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 17/24, 6/24, 0/24]，權重總和恰為一。令 Z=X+(3)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"867/16","explanation_zh":"變異數=867/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057041,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"c":3,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"3/4\"],[\"17\",\"1/4\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0042","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 6), (1, 6), (2, 24), (3, 1015)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 18/22, 3/22, 0/22]，權重總和恰為一。令 Z=X+(4)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"4617/121","explanation_zh":"變異數=4617/121。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057042,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"c":4,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"19/22\"],[\"18\",\"3/22\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0043","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, -4), (3, 1016)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 19/24, 4/24, 0/24]，權重總和恰為一。令 Z=X+(5)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1805/36","explanation_zh":"變異數=1805/36。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057043,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"c":5,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"19\",\"1/6\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0044","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 8), (1, 8), (2, 28), (3, 1017)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 20/26, 5/26, 0/26]，權重總和恰為一。令 Z=X+(6)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"10500/169","explanation_zh":"變異數=10500/169。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057044,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"c":6,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"21/26\"],[\"20\",\"5/26\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0045","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -25), (1, -25), (2, -4), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 21/28, 6/28, 0/28]，權重總和恰為一。令 Z=X+(7)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"297/4","explanation_zh":"變異數=297/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057045,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"c":7,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"11/14\"],[\"21\",\"3/14\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0046","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 10), (1, 10), (2, 32), (3, 1019)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 22/26, 3/26, 0/26]，權重總和恰為一。令 Z=X+(8)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"8349/169","explanation_zh":"變異數=8349/169。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057046,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"c":8,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"23/26\"],[\"22\",\"3/26\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0047","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -27), (1, -27), (2, -4), (3, 1020)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 23/28, 4/28, 0/28]，權重總和恰為一。令 Z=X+(9)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"3174/49","explanation_zh":"變異數=3174/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057047,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"c":9,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"6/7\"],[\"23\",\"1/7\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0048","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 12), (1, 12), (2, 36), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 24/30, 5/30, 0/30]，權重總和恰為一。令 Z=X+(10)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"80/1","explanation_zh":"變異數=80/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057048,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"c":10,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"24\",\"1/6\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0049","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -29), (1, -29), (2, -4), (3, 1022)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/32, 25/32, 6/32, 0/32]，權重總和恰為一。令 Z=X+(11)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"24375/256","explanation_zh":"變異數=24375/256。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057049,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"c":11,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/16\"],[\"25\",\"3/16\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0050","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"平移後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 14), (1, 14), (2, 40), (3, 1023)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 26/30, 3/30, 0/30]，權重總和恰為一。令 Z=X+(12)，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1521/25","explanation_zh":"變異數=1521/25。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["平移後的機率變異數"]},"variant":{"family_id":"qf-variance-shift","seed":202610057050,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"c":12,"operation":"shift_variance","semantic_template":"qf-variance-shift/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"9/10\"],[\"26\",\"1/10\"]],\"operation\":\"shift_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0051","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 5), (1, 5), (2, 999)}，各結果依列序 [0, 1, 2] 的非負機率權重為 [1/3, 2/3, 0/3]，權重總和恰為一。令 Z=0X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"0/1","explanation_zh":"變異數=0/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057051,"parameters":{"S":[0,1,2],"F":"powerset(S)","values":[5,5,999],"weights":[1,2,0],"denominator":3,"k":0,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"k\":0,\"normalized_model\":true,\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0052","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, -4), (3, 1000)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/8, 3/8, 4/8, 0/8]，權重總和恰為一。令 Z=-2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"9/1","explanation_zh":"變異數=9/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057052,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"k":-2,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"1/2\"],[\"3\",\"1/2\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0053","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, -4), (3, 1001)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 4/10, 5/10, 0/10]，權重總和恰為一。令 Z=3X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"36/1","explanation_zh":"變異數=36/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057053,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"k":3,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":3,\"law\":[[\"0\",\"1/2\"],[\"4\",\"1/2\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0054","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, -4), (3, 1002)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 5/12, 6/12, 0/12]，權重總和恰為一。令 Z=-4X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"100/1","explanation_zh":"變異數=100/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057054,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"k":-4,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":4,\"law\":[[\"0\",\"1/2\"],[\"5\",\"1/2\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0055","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 0), (3, 1003)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 6/10, 3/10, 0/10]，權重總和恰為一。令 Z=5X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"189/1","explanation_zh":"變異數=189/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057055,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"k":5,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":5,\"law\":[[\"0\",\"7/10\"],[\"6\",\"3/10\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0056","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。令 Z=-1X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"98/9","explanation_zh":"變異數=98/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057056,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"k":-1,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":1,\"law\":[[\"0\",\"2/3\"],[\"7\",\"1/3\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0057","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 4), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 8/14, 5/14, 0/14]，權重總和恰為一。令 Z=2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"2880/49","explanation_zh":"變異數=2880/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057057,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"k":2,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"9/14\"],[\"8\",\"5/14\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0058","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, -4), (3, 1006)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 9/16, 6/16, 0/16]，權重總和恰為一。令 Z=-3X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"10935/64","explanation_zh":"變異數=10935/64。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057058,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"k":-3,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":3,\"law\":[[\"0\",\"5/8\"],[\"9\",\"3/8\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0059","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 8), (3, 1007)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 10/14, 3/14, 0/14]，權重總和恰為一。令 Z=4X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"13200/49","explanation_zh":"變異數=13200/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057059,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"k":4,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":4,\"law\":[[\"0\",\"11/14\"],[\"10\",\"3/14\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0060","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, -4), (3, 1008)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 11/16, 4/16, 0/16]，權重總和恰為一。令 Z=-5X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"9075/16","explanation_zh":"變異數=9075/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057060,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"k":-5,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":5,\"law\":[[\"0\",\"3/4\"],[\"11\",\"1/4\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0061","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0), (1, 0), (2, 12), (3, 1009)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 12/18, 5/18, 0/18]，權重總和恰為一。令 Z=1X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"260/9","explanation_zh":"變異數=260/9。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057061,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"k":1,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":1,\"law\":[[\"0\",\"13/18\"],[\"12\",\"5/18\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0062","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, -4), (3, 1010)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 13/20, 6/20, 0/20]，權重總和恰為一。令 Z=-2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"3549/25","explanation_zh":"變異數=3549/25。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057062,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"k":-2,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"7/10\"],[\"13\",\"3/10\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0063","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 2), (1, 2), (2, 16), (3, 1011)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 14/18, 3/18, 0/18]，權重總和恰為一。令 Z=3X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"245/1","explanation_zh":"變異數=245/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057063,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"k":3,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":3,\"law\":[[\"0\",\"5/6\"],[\"14\",\"1/6\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0064","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, -4), (3, 1012)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 15/20, 4/20, 0/20]，權重總和恰為一。令 Z=-4X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"576/1","explanation_zh":"變異數=576/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057064,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"k":-4,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":4,\"law\":[[\"0\",\"4/5\"],[\"15\",\"1/5\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0065","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 4), (1, 4), (2, 20), (3, 1013)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 16/22, 5/22, 0/22]，權重總和恰為一。令 Z=5X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"136000/121","explanation_zh":"變異數=136000/121。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057065,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"k":5,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":5,\"law\":[[\"0\",\"17/22\"],[\"16\",\"5/22\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0066","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, -4), (3, 1014)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 17/24, 6/24, 0/24]，權重總和恰為一。令 Z=-1X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"867/16","explanation_zh":"變異數=867/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057066,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"k":-1,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":1,\"law\":[[\"0\",\"3/4\"],[\"17\",\"1/4\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0067","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 6), (1, 6), (2, 24), (3, 1015)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 18/22, 3/22, 0/22]，權重總和恰為一。令 Z=2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"18468/121","explanation_zh":"變異數=18468/121。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057067,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"k":2,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"19/22\"],[\"18\",\"3/22\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0068","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, -4), (3, 1016)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 19/24, 4/24, 0/24]，權重總和恰為一。令 Z=-3X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1805/4","explanation_zh":"變異數=1805/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057068,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"k":-3,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":3,\"law\":[[\"0\",\"5/6\"],[\"19\",\"1/6\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0069","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 8), (1, 8), (2, 28), (3, 1017)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 20/26, 5/26, 0/26]，權重總和恰為一。令 Z=4X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"168000/169","explanation_zh":"變異數=168000/169。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057069,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"k":4,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":4,\"law\":[[\"0\",\"21/26\"],[\"20\",\"5/26\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0070","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -25), (1, -25), (2, -4), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 21/28, 6/28, 0/28]，權重總和恰為一。令 Z=-5X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"7425/4","explanation_zh":"變異數=7425/4。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057070,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"k":-5,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":5,\"law\":[[\"0\",\"11/14\"],[\"21\",\"3/14\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0071","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 10), (1, 10), (2, 32), (3, 1019)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 22/26, 3/26, 0/26]，權重總和恰為一。令 Z=1X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"8349/169","explanation_zh":"變異數=8349/169。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057071,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"k":1,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":1,\"law\":[[\"0\",\"23/26\"],[\"22\",\"3/26\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0072","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -27), (1, -27), (2, -4), (3, 1020)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 23/28, 4/28, 0/28]，權重總和恰為一。令 Z=-2X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"12696/49","explanation_zh":"變異數=12696/49。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057072,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"k":-2,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":2,\"law\":[[\"0\",\"6/7\"],[\"23\",\"1/7\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0073","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 12), (1, 12), (2, 36), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 24/30, 5/30, 0/30]，權重總和恰為一。令 Z=3X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"720/1","explanation_zh":"變異數=720/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057073,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"k":3,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":3,\"law\":[[\"0\",\"5/6\"],[\"24\",\"1/6\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0074","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -29), (1, -29), (2, -4), (3, 1022)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/32, 25/32, 6/32, 0/32]，權重總和恰為一。令 Z=-4X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"24375/16","explanation_zh":"變異數=24375/16。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057074,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"k":-4,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":4,\"law\":[[\"0\",\"13/16\"],[\"25\",\"3/16\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0075","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"縮放後的機率變異數","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 14), (1, 14), (2, 40), (3, 1023)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 26/30, 3/30, 0/30]，權重總和恰為一。令 Z=5X，求精確 Var(Z)=E[(Z−E[Z])²]；不是 n−1 分母的樣本變異數。","answer":{"kind":"expression","value":"1521/1","explanation_zh":"變異數=1521/1。產生端用正規化權重的兩兩差平方計算，Σ_(i<j)w_iw_j(z_i−z_j)²；平移差值不變，縮放平方帶來 k²。零權重不貢獻，這不是樣本變異數。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["縮放後的機率變異數"]},"variant":{"family_id":"qf-variance-scale","seed":202610057075,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"k":5,"operation":"scale_variance","semantic_template":"qf-variance-scale/{\"F\":\"powerset(S)\",\"k\":5,\"law\":[[\"0\",\"9/10\"],[\"26\",\"1/10\"]],\"operation\":\"scale_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0076","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 5), (1, 5), (2, 999)}，各結果依列序 [0, 1, 2] 的非負機率權重為 [1/3, 2/3, 0/3]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"是","explanation_zh":"正權重取值={5}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057076,"parameters":{"S":[0,1,2],"F":"powerset(S)","values":[5,5,999],"weights":[1,2,0],"denominator":3,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/1\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0077","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -7), (1, -7), (2, -4), (3, 1000)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/8, 3/8, 4/8, 0/8]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-7, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057077,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"3\",\"1/2\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0078","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -8), (1, -8), (2, -4), (3, 1001)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 4/10, 5/10, 0/10]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-8, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057078,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"4\",\"1/2\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0079","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -9), (1, -9), (2, -4), (3, 1002)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 5/12, 6/12, 0/12]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-9, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057079,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"1/2\"],[\"5\",\"1/2\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0080","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -6), (1, -6), (2, 0), (3, 1003)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/10, 6/10, 3/10, 0/10]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-6, 0}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057080,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"7/10\"],[\"6\",\"3/10\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0081","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -11), (1, -11), (2, -4), (3, 1004)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/12, 7/12, 4/12, 0/12]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-11, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057081,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"2/3\"],[\"7\",\"1/3\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0082","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -4), (1, -4), (2, 4), (3, 1005)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 8/14, 5/14, 0/14]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-4, 4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057082,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"9/14\"],[\"8\",\"5/14\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0083","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -13), (1, -13), (2, -4), (3, 1006)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 9/16, 6/16, 0/16]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-13, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057083,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/8\"],[\"9\",\"3/8\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0084","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -2), (1, -2), (2, 8), (3, 1007)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/14, 10/14, 3/14, 0/14]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-2, 8}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057084,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"11/14\"],[\"10\",\"3/14\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0085","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -15), (1, -15), (2, -4), (3, 1008)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/16, 11/16, 4/16, 0/16]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-15, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057085,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"3/4\"],[\"11\",\"1/4\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0086","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 0), (1, 0), (2, 12), (3, 1009)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 12/18, 5/18, 0/18]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={0, 12}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057086,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/18\"],[\"12\",\"5/18\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0087","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -17), (1, -17), (2, -4), (3, 1010)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 13/20, 6/20, 0/20]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-17, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057087,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"7/10\"],[\"13\",\"3/10\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0088","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 2), (1, 2), (2, 16), (3, 1011)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/18, 14/18, 3/18, 0/18]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={2, 16}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057088,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"14\",\"1/6\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0089","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -19), (1, -19), (2, -4), (3, 1012)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/20, 15/20, 4/20, 0/20]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-19, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057089,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"4/5\"],[\"15\",\"1/5\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0090","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 4), (1, 4), (2, 20), (3, 1013)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 16/22, 5/22, 0/22]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={4, 20}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057090,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"17/22\"],[\"16\",\"5/22\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0091","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -21), (1, -21), (2, -4), (3, 1014)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 17/24, 6/24, 0/24]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-21, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057091,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"3/4\"],[\"17\",\"1/4\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0092","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 6), (1, 6), (2, 24), (3, 1015)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/22, 18/22, 3/22, 0/22]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={6, 24}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057092,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"19/22\"],[\"18\",\"3/22\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0093","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -23), (1, -23), (2, -4), (3, 1016)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/24, 19/24, 4/24, 0/24]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-23, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057093,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"19\",\"1/6\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0094","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 8), (1, 8), (2, 28), (3, 1017)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 20/26, 5/26, 0/26]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={8, 28}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057094,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"21/26\"],[\"20\",\"5/26\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0095","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -25), (1, -25), (2, -4), (3, 1018)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 21/28, 6/28, 0/28]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-25, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057095,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"11/14\"],[\"21\",\"3/14\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0096","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 10), (1, 10), (2, 32), (3, 1019)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/26, 22/26, 3/26, 0/26]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={10, 32}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057096,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"23/26\"],[\"22\",\"3/26\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0097","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -27), (1, -27), (2, -4), (3, 1020)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/28, 23/28, 4/28, 0/28]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-27, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057097,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"6/7\"],[\"23\",\"1/7\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0098","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 12), (1, 12), (2, 36), (3, 1021)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 24/30, 5/30, 0/30]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={12, 36}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057098,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"5/6\"],[\"24\",\"1/6\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0099","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, -29), (1, -29), (2, -4), (3, 1022)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/32, 25/32, 6/32, 0/32]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={-29, -4}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057099,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/16\"],[\"25\",\"3/16\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r07-variance-0100","batch_id":"batch-20261004-advance-20261005-r07-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-variance"],"title_zh":"正機率支持上的零變異數判定","stem_zh":"非空有限 S={0, 1, 2, 3}、事件族 F=𝒫(S)，總函數 X:S→R 的值表為 {(0, 14), (1, 14), (2, 40), (3, 1023)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/30, 26/30, 3/30, 0/30]，權重總和恰為一。依此機率分布，Var(X) 是否為零？只能要求正權重結果上的值相同。","answer":{"kind":"short_text","value":"否","explanation_zh":"正權重取值={14, 40}。正權重支持上常值才有零變異數；零權重結果可有不同值，這不是樣本變異數，也不要求全 S 上常值。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["正機率支持上的零變異數判定"]},"variant":{"family_id":"qf-variance-zero-support","seed":202610057100,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"operation":"zero_variance","semantic_template":"qf-variance-zero-support/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"9/10\"],[\"26\",\"1/10\"]],\"operation\":\"zero_variance\"}"}},"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":"Codex independent producer (inherited model; exact runtime model not exposed)","generated_at":"2026-10-04","contract_version":"ocme-question-factory-v0.1"}}