{"schema_version":"ocme-question-v0.1","id":"q-20261004-advance-20261005-r06-expectation-0001","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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, -8), (3, 999)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/6, 2/6, 3/6, 0/6]，權重總和恰為一。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"期望=-9/1。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056001,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,-8,999],"weights":[1,2,3,0],"denominator":6,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-10\",\"1/2\"],[\"-8\",\"1/2\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0002","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-11/2","explanation_zh":"期望=-11/2。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056002,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-7\",\"1/2\"],[\"-4\",\"1/2\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0003","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-6/1","explanation_zh":"期望=-6/1。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056003,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-8\",\"1/2\"],[\"-4\",\"1/2\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0004","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-13/2","explanation_zh":"期望=-13/2。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056004,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-9\",\"1/2\"],[\"-4\",\"1/2\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0005","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-21/5","explanation_zh":"期望=-21/5。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056005,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-6\",\"7/10\"],[\"0\",\"3/10\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0006","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-26/3","explanation_zh":"期望=-26/3。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056006,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-11\",\"2/3\"],[\"-4\",\"1/3\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0007","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-8/7","explanation_zh":"期望=-8/7。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056007,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-4\",\"9/14\"],[\"4\",\"5/14\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0008","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-77/8","explanation_zh":"期望=-77/8。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056008,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-13\",\"5/8\"],[\"-4\",\"3/8\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0009","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"1/7","explanation_zh":"期望=1/7。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056009,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-2\",\"11/14\"],[\"8\",\"3/14\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0010","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-49/4","explanation_zh":"期望=-49/4。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056010,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-15\",\"3/4\"],[\"-4\",\"1/4\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0011","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"10/3","explanation_zh":"期望=10/3。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056011,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"13/18\"],[\"12\",\"5/18\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0012","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-131/10","explanation_zh":"期望=-131/10。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056012,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-17\",\"7/10\"],[\"-4\",\"3/10\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0013","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"13/3","explanation_zh":"期望=13/3。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056013,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"2\",\"5/6\"],[\"16\",\"1/6\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0014","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-16/1","explanation_zh":"期望=-16/1。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056014,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-19\",\"4/5\"],[\"-4\",\"1/5\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0015","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"84/11","explanation_zh":"期望=84/11。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056015,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"4\",\"17/22\"],[\"20\",\"5/22\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0016","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-67/4","explanation_zh":"期望=-67/4。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056016,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-21\",\"3/4\"],[\"-4\",\"1/4\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0017","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"93/11","explanation_zh":"期望=93/11。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056017,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"6\",\"19/22\"],[\"24\",\"3/22\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0018","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-119/6","explanation_zh":"期望=-119/6。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056018,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-23\",\"5/6\"],[\"-4\",\"1/6\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0019","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"154/13","explanation_zh":"期望=154/13。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056019,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"8\",\"21/26\"],[\"28\",\"5/26\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0020","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-41/2","explanation_zh":"期望=-41/2。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056020,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-25\",\"11/14\"],[\"-4\",\"3/14\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0021","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"163/13","explanation_zh":"期望=163/13。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056021,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"10\",\"23/26\"],[\"32\",\"3/26\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0022","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-166/7","explanation_zh":"期望=-166/7。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056022,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-27\",\"6/7\"],[\"-4\",\"1/7\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0023","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"16/1","explanation_zh":"期望=16/1。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056023,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"12\",\"5/6\"],[\"36\",\"1/6\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0024","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"-389/16","explanation_zh":"期望=-389/16。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056024,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"-29\",\"13/16\"],[\"-4\",\"3/16\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0025","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"非等可能有限模型的期望","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。求精確 E[X]，不可改成不同值的普通平均。","answer":{"kind":"expression","value":"83/5","explanation_zh":"期望=83/5。每個結果的值依給定權重加權；期望可以為負或不在實際值域中，不是樣本均值估計。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["非等可能有限模型的期望"]},"variant":{"family_id":"qf-expectation-weighted","seed":202610056025,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"operation":"expectation","semantic_template":"qf-expectation-weighted/{\"F\":\"powerset(S)\",\"law\":[[\"14\",\"9/10\"],[\"40\",\"1/10\"]],\"operation\":\"expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0026","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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, -8), (3, 999)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/6, 2/6, 3/6, 0/6]，權重總和恰為一。令 Z=0X+(-5)，求精確 E[Z]。","answer":{"kind":"expression","value":"-5/1","explanation_zh":"期望=-5/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056026,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,-8,999],"weights":[1,2,3,0],"denominator":6,"a":0,"b":-5,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"a\":0,\"b\":-5,\"normalized_model\":true,\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0027","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(-4)，求精確 E[Z]。","answer":{"kind":"expression","value":"7/1","explanation_zh":"期望=7/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056027,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"a":-2,"b":-4,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":-4,\"law\":[[\"-7\",\"1/2\"],[\"-4\",\"1/2\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0028","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(-3)，求精確 E[Z]。","answer":{"kind":"expression","value":"-21/1","explanation_zh":"期望=-21/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056028,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"a":3,"b":-3,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":3,\"b\":-3,\"law\":[[\"-8\",\"1/2\"],[\"-4\",\"1/2\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0029","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(-2)，求精確 E[Z]。","answer":{"kind":"expression","value":"24/1","explanation_zh":"期望=24/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056029,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"a":-4,"b":-2,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-4,\"b\":-2,\"law\":[[\"-9\",\"1/2\"],[\"-4\",\"1/2\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0030","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=1X+(-1)，求精確 E[Z]。","answer":{"kind":"expression","value":"-26/5","explanation_zh":"期望=-26/5。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056030,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"a":1,"b":-1,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":1,\"b\":-1,\"law\":[[\"-6\",\"7/10\"],[\"0\",\"3/10\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0031","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-2X+(0)，求精確 E[Z]。","answer":{"kind":"expression","value":"52/3","explanation_zh":"期望=52/3。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056031,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"a":-2,"b":0,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":0,\"law\":[[\"-11\",\"2/3\"],[\"-4\",\"1/3\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0032","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=3X+(1)，求精確 E[Z]。","answer":{"kind":"expression","value":"-17/7","explanation_zh":"期望=-17/7。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056032,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"a":3,"b":1,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":3,\"b\":1,\"law\":[[\"-4\",\"9/14\"],[\"4\",\"5/14\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0033","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-4X+(2)，求精確 E[Z]。","answer":{"kind":"expression","value":"81/2","explanation_zh":"期望=81/2。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056033,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"a":-4,"b":2,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-4,\"b\":2,\"law\":[[\"-13\",\"5/8\"],[\"-4\",\"3/8\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0034","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=1X+(3)，求精確 E[Z]。","answer":{"kind":"expression","value":"22/7","explanation_zh":"期望=22/7。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056034,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"a":1,"b":3,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":1,\"b\":3,\"law\":[[\"-2\",\"11/14\"],[\"8\",\"3/14\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0035","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-2X+(4)，求精確 E[Z]。","answer":{"kind":"expression","value":"57/2","explanation_zh":"期望=57/2。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056035,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"a":-2,"b":4,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":4,\"law\":[[\"-15\",\"3/4\"],[\"-4\",\"1/4\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0036","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=3X+(5)，求精確 E[Z]。","answer":{"kind":"expression","value":"15/1","explanation_zh":"期望=15/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056036,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"a":3,"b":5,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":3,\"b\":5,\"law\":[[\"0\",\"13/18\"],[\"12\",\"5/18\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0037","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-4X+(6)，求精確 E[Z]。","answer":{"kind":"expression","value":"292/5","explanation_zh":"期望=292/5。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056037,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"a":-4,"b":6,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-4,\"b\":6,\"law\":[[\"-17\",\"7/10\"],[\"-4\",\"3/10\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0038","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=1X+(7)，求精確 E[Z]。","answer":{"kind":"expression","value":"34/3","explanation_zh":"期望=34/3。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056038,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"a":1,"b":7,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":1,\"b\":7,\"law\":[[\"2\",\"5/6\"],[\"16\",\"1/6\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0039","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-2X+(8)，求精確 E[Z]。","answer":{"kind":"expression","value":"40/1","explanation_zh":"期望=40/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056039,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"a":-2,"b":8,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":8,\"law\":[[\"-19\",\"4/5\"],[\"-4\",\"1/5\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0040","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=3X+(9)，求精確 E[Z]。","answer":{"kind":"expression","value":"351/11","explanation_zh":"期望=351/11。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056040,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"a":3,"b":9,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":3,\"b\":9,\"law\":[[\"4\",\"17/22\"],[\"20\",\"5/22\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0041","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-4X+(10)，求精確 E[Z]。","answer":{"kind":"expression","value":"77/1","explanation_zh":"期望=77/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056041,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"a":-4,"b":10,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-4,\"b\":10,\"law\":[[\"-21\",\"3/4\"],[\"-4\",\"1/4\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0042","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=1X+(11)，求精確 E[Z]。","answer":{"kind":"expression","value":"214/11","explanation_zh":"期望=214/11。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056042,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"a":1,"b":11,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":1,\"b\":11,\"law\":[[\"6\",\"19/22\"],[\"24\",\"3/22\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0043","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-2X+(12)，求精確 E[Z]。","answer":{"kind":"expression","value":"155/3","explanation_zh":"期望=155/3。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056043,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"a":-2,"b":12,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":12,\"law\":[[\"-23\",\"5/6\"],[\"-4\",\"1/6\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0044","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=3X+(13)，求精確 E[Z]。","answer":{"kind":"expression","value":"631/13","explanation_zh":"期望=631/13。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056044,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"a":3,"b":13,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":3,\"b\":13,\"law\":[[\"8\",\"21/26\"],[\"28\",\"5/26\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0045","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=-4X+(14)，求精確 E[Z]。","answer":{"kind":"expression","value":"96/1","explanation_zh":"期望=96/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056045,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"a":-4,"b":14,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-4,\"b\":14,\"law\":[[\"-25\",\"11/14\"],[\"-4\",\"3/14\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0046","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(15)，求精確 E[Z]。","answer":{"kind":"expression","value":"358/13","explanation_zh":"期望=358/13。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056046,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"a":1,"b":15,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":1,\"b\":15,\"law\":[[\"10\",\"23/26\"],[\"32\",\"3/26\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0047","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(16)，求精確 E[Z]。","answer":{"kind":"expression","value":"444/7","explanation_zh":"期望=444/7。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056047,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"a":-2,"b":16,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-2,\"b\":16,\"law\":[[\"-27\",\"6/7\"],[\"-4\",\"1/7\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0048","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(17)，求精確 E[Z]。","answer":{"kind":"expression","value":"65/1","explanation_zh":"期望=65/1。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056048,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"a":3,"b":17,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":3,\"b\":17,\"law\":[[\"12\",\"5/6\"],[\"36\",\"1/6\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0049","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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+(18)，求精確 E[Z]。","answer":{"kind":"expression","value":"461/4","explanation_zh":"期望=461/4。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056049,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"a":-4,"b":18,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":-4,\"b\":18,\"law\":[[\"-29\",\"13/16\"],[\"-4\",\"3/16\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0050","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"仿射變換的有限期望","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=1X+(19)，求精確 E[Z]。","answer":{"kind":"expression","value":"178/5","explanation_zh":"期望=178/5。本產生端先逐結果形成 Z 再加權，與 aE[X]+b 相容；a=0 時只剩常數 b，不使用浮點近似。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["仿射變換的有限期望"]},"variant":{"family_id":"qf-expectation-affine","seed":202610056050,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"a":1,"b":19,"operation":"affine_expectation","semantic_template":"qf-expectation-affine/{\"F\":\"powerset(S)\",\"a\":1,\"b\":19,\"law\":[[\"14\",\"9/10\"],[\"40\",\"1/10\"]],\"operation\":\"affine_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0051","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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, -8), (3, 999)}，各結果依列序 [0, 1, 2, 3] 的非負機率權重為 [1/6, 2/6, 3/6, 0/6]，權重總和恰為一。同一 S 上 Y 的值表為 {(0, -20), (1, -20), (2, -16), (3, 1998)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-27/1","explanation_zh":"期望=-27/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056051,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-10,-10,-8,999],"weights":[1,2,3,0],"denominator":6,"Y":[-20,-20,-16,1998],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-10\",\"-20\",\"1/2\"],[\"-8\",\"-16\",\"1/2\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0052","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -13), (1, -13), (2, -7), (3, 2001)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-31/2","explanation_zh":"期望=-31/2。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056052,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-7,-7,-4,1000],"weights":[1,3,4,0],"denominator":8,"Y":[-13,-13,-7,2001],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-7\",\"-13\",\"1/2\"],[\"-4\",\"-7\",\"1/2\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0053","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -14), (1, -14), (2, -6), (3, 2004)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-16/1","explanation_zh":"期望=-16/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056053,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-8,-8,-4,1001],"weights":[1,4,5,0],"denominator":10,"Y":[-14,-14,-6,2004],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-8\",\"-14\",\"1/2\"],[\"-4\",\"-6\",\"1/2\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0054","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -15), (1, -15), (2, -5), (3, 2007)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-33/2","explanation_zh":"期望=-33/2。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056054,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-9,-9,-4,1002],"weights":[1,5,6,0],"denominator":12,"Y":[-15,-15,-5,2007],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-9\",\"-15\",\"1/2\"],[\"-4\",\"-5\",\"1/2\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0055","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -8), (1, -8), (2, 4), (3, 2010)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-43/5","explanation_zh":"期望=-43/5。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056055,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-6,-6,0,1003],"weights":[1,6,3,0],"denominator":10,"Y":[-8,-8,4,2010],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-6\",\"-8\",\"7/10\"],[\"0\",\"4\",\"3/10\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0056","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -22), (1, -22), (2, -8), (3, 2008)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-26/1","explanation_zh":"期望=-26/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056056,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-11,-11,-4,1004],"weights":[1,7,4,0],"denominator":12,"Y":[-22,-22,-8,2008],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-11\",\"-22\",\"2/3\"],[\"-4\",\"-8\",\"1/3\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0057","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -7), (1, -7), (2, 9), (3, 2011)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-17/7","explanation_zh":"期望=-17/7。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056057,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-4,-4,4,1005],"weights":[1,8,5,0],"denominator":14,"Y":[-7,-7,9,2011],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-4\",\"-7\",\"9/14\"],[\"4\",\"9\",\"5/14\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0058","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -24), (1, -24), (2, -6), (3, 2014)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-215/8","explanation_zh":"期望=-215/8。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056058,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-13,-13,-4,1006],"weights":[1,9,6,0],"denominator":16,"Y":[-24,-24,-6,2014],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-13\",\"-24\",\"5/8\"],[\"-4\",\"-6\",\"3/8\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0059","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -1), (1, -1), (2, 19), (3, 2017)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"24/7","explanation_zh":"期望=24/7。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056059,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-2,-2,8,1007],"weights":[1,10,3,0],"denominator":14,"Y":[-1,-1,19,2017],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-2\",\"-1\",\"11/14\"],[\"8\",\"19\",\"3/14\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0060","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -26), (1, -26), (2, -4), (3, 2020)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-131/4","explanation_zh":"期望=-131/4。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056060,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-15,-15,-4,1008],"weights":[1,11,4,0],"denominator":16,"Y":[-26,-26,-4,2020],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-15\",\"-26\",\"3/4\"],[\"-4\",\"-4\",\"1/4\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0061","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 0), (1, 0), (2, 24), (3, 2018)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"10/1","explanation_zh":"期望=10/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056061,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[0,0,12,1009],"weights":[1,12,5,0],"denominator":18,"Y":[0,0,24,2018],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"0\",\"0\",\"13/18\"],[\"12\",\"24\",\"5/18\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0062","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -33), (1, -33), (2, -7), (3, 2021)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-383/10","explanation_zh":"期望=-383/10。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056062,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-17,-17,-4,1010],"weights":[1,13,6,0],"denominator":20,"Y":[-33,-33,-7,2021],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-17\",\"-33\",\"7/10\"],[\"-4\",\"-7\",\"3/10\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0063","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 6), (1, 6), (2, 34), (3, 2024)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"15/1","explanation_zh":"期望=15/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056063,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[2,2,16,1011],"weights":[1,14,3,0],"denominator":18,"Y":[6,6,34,2024],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"2\",\"6\",\"5/6\"],[\"16\",\"34\",\"1/6\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0064","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -35), (1, -35), (2, -5), (3, 2027)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-45/1","explanation_zh":"期望=-45/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056064,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-19,-19,-4,1012],"weights":[1,15,4,0],"denominator":20,"Y":[-35,-35,-5,2027],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-19\",\"-35\",\"4/5\"],[\"-4\",\"-5\",\"1/5\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0065","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 12), (1, 12), (2, 44), (3, 2030)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"296/11","explanation_zh":"期望=296/11。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056065,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[4,4,20,1013],"weights":[1,16,5,0],"denominator":22,"Y":[12,12,44,2030],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"4\",\"12\",\"17/22\"],[\"20\",\"44\",\"5/22\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0066","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -42), (1, -42), (2, -8), (3, 2028)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-201/4","explanation_zh":"期望=-201/4。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056066,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-21,-21,-4,1014],"weights":[1,17,6,0],"denominator":24,"Y":[-42,-42,-8,2028],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-21\",\"-42\",\"3/4\"],[\"-4\",\"-8\",\"1/4\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0067","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 13), (1, 13), (2, 49), (3, 2031)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"290/11","explanation_zh":"期望=290/11。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056067,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[6,6,24,1015],"weights":[1,18,3,0],"denominator":22,"Y":[13,13,49,2031],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"6\",\"13\",\"19/22\"],[\"24\",\"49\",\"3/22\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0068","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -44), (1, -44), (2, -6), (3, 2034)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-115/2","explanation_zh":"期望=-115/2。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056068,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-23,-23,-4,1016],"weights":[1,19,4,0],"denominator":24,"Y":[-44,-44,-6,2034],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-23\",\"-44\",\"5/6\"],[\"-4\",\"-6\",\"1/6\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0069","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 19), (1, 19), (2, 59), (3, 2037)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"501/13","explanation_zh":"期望=501/13。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056069,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[8,8,28,1017],"weights":[1,20,5,0],"denominator":26,"Y":[19,19,59,2037],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"8\",\"19\",\"21/26\"],[\"28\",\"59\",\"5/26\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0070","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -46), (1, -46), (2, -4), (3, 2040)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-115/2","explanation_zh":"期望=-115/2。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056070,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-25,-25,-4,1018],"weights":[1,21,6,0],"denominator":28,"Y":[-46,-46,-4,2040],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-25\",\"-46\",\"11/14\"],[\"-4\",\"-4\",\"3/14\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0071","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 20), (1, 20), (2, 64), (3, 2038)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"489/13","explanation_zh":"期望=489/13。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056071,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[10,10,32,1019],"weights":[1,22,3,0],"denominator":26,"Y":[20,20,64,2038],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"10\",\"20\",\"23/26\"],[\"32\",\"64\",\"3/26\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0072","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -53), (1, -53), (2, -7), (3, 2041)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-491/7","explanation_zh":"期望=-491/7。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056072,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-27,-27,-4,1020],"weights":[1,23,4,0],"denominator":28,"Y":[-53,-53,-7,2041],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-27\",\"-53\",\"6/7\"],[\"-4\",\"-7\",\"1/7\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0073","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 26), (1, 26), (2, 74), (3, 2044)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"50/1","explanation_zh":"期望=50/1。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056073,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[12,12,36,1021],"weights":[1,24,5,0],"denominator":30,"Y":[26,26,74,2044],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"12\",\"26\",\"5/6\"],[\"36\",\"74\",\"1/6\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0074","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, -55), (1, -55), (2, -5), (3, 2047)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"-1119/16","explanation_zh":"期望=-1119/16。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056074,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[-29,-29,-4,1022],"weights":[1,25,6,0],"denominator":32,"Y":[-55,-55,-5,2047],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"-29\",\"-55\",\"13/16\"],[\"-4\",\"-5\",\"3/16\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0075","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"共用模型兩變數和的期望","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。同一 S 上 Y 的值表為 {(0, 32), (1, 32), (2, 84), (3, 2050)}；求精確 E[X+Y]，不另造獨立模型。","answer":{"kind":"expression","value":"269/5","explanation_zh":"期望=269/5。在每個相同結果先相加再加權；E[X+Y] 的線性性不要求獨立，也不推出乘積期望公式。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["共用模型兩變數和的期望"]},"variant":{"family_id":"qf-expectation-same-model-sum","seed":202610056075,"parameters":{"S":[0,1,2,3],"F":"powerset(S)","values":[14,14,40,1023],"weights":[1,26,3,0],"denominator":30,"Y":[32,32,84,2050],"operation":"sum_expectation","semantic_template":"qf-expectation-same-model-sum/{\"F\":\"powerset(S)\",\"law\":[[\"14\",\"32\",\"9/10\"],[\"40\",\"84\",\"1/10\"]],\"operation\":\"sum_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0076","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-12 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-12/1","explanation_zh":"E[c]=-12/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056076,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-12,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-12,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0077","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-11 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-11/1","explanation_zh":"E[c]=-11/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056077,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-11,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-11,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0078","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-10 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-10/1","explanation_zh":"E[c]=-10/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056078,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-10,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-10,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0079","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-9 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-9/1","explanation_zh":"E[c]=-9/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056079,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-9,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-9,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0080","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-8 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-8/1","explanation_zh":"E[c]=-8/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056080,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-8,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-8,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0081","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-7 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-7/1","explanation_zh":"E[c]=-7/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056081,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-7,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-7,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0082","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-6 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-6/1","explanation_zh":"E[c]=-6/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056082,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-6,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-6,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0083","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-5 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-5/1","explanation_zh":"E[c]=-5/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056083,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-5,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-5,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0084","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-4 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-4/1","explanation_zh":"E[c]=-4/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056084,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-4,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-4,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0085","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-3 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-3/1","explanation_zh":"E[c]=-3/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056085,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-3,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-3,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0086","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-2 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-2/1","explanation_zh":"E[c]=-2/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056086,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-2,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-2,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0087","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=-1 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"-1/1","explanation_zh":"E[c]=-1/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056087,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":-1,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":-1,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0088","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=0 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"0/1","explanation_zh":"E[c]=0/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056088,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":0,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":0,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0089","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=1 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"1/1","explanation_zh":"E[c]=1/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056089,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":1,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":1,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0090","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=2 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"2/1","explanation_zh":"E[c]=2/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056090,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":2,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":2,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0091","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=3 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"3/1","explanation_zh":"E[c]=3/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056091,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":3,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":3,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0092","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=4 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"4/1","explanation_zh":"E[c]=4/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056092,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":4,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":4,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0093","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=5 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"5/1","explanation_zh":"E[c]=5/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056093,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":5,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":5,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0094","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=6 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"6/1","explanation_zh":"E[c]=6/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056094,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":6,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":6,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0095","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=7 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"7/1","explanation_zh":"E[c]=7/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056095,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":7,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":7,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0096","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=8 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"8/1","explanation_zh":"E[c]=8/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056096,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":8,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":8,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0097","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=9 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"9/1","explanation_zh":"E[c]=9/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056097,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":9,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":9,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0098","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=10 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"10/1","explanation_zh":"E[c]=10/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056098,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":10,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":10,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0099","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=11 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"11/1","explanation_zh":"E[c]=11/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056099,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":11,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":11,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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-r06-expectation-0100","batch_id":"batch-20261004-advance-20261005-r06-expectation-foundations","language":"zh-Hant","source_mko_ids":["mko-expectation"],"title_zh":"歸一化模型中常數的期望","stem_zh":"非空有限 S={0, 1}、事件族 F=𝒫(S)，各結果依列序 [0, 1] 的非負機率權重為 [1/4, 3/4]，權重總和恰為一。令 X(s)=12 對每個 s∈S 成立，求精確 E[X]。","answer":{"kind":"expression","value":"12/1","explanation_zh":"E[c]=12/1。常數乘權重總和一仍為該常數；不同機率配置不創造不同的常數期望問題。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["歸一化模型中常數的期望"]},"variant":{"family_id":"qf-expectation-constant","seed":202610056100,"parameters":{"S":[0,1],"F":"powerset(S)","weights":[1,3],"denominator":4,"c":12,"operation":"constant_expectation","semantic_template":"qf-expectation-constant/{\"c\":12,\"normalized_model\":true,\"operation\":\"constant_expectation\"}"}},"verification":{"status":"candidate","checks":["producer_exact_integer_rational_arithmetic","producer_finite_enumeration","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"僅為產生端候選及自檢；主端須另做獨立驗收，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"}}