{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0001","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 1, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 2, 1, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064001,"parameters":{"laws":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],"statistic":[0,1,2,3],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"parameter\":0,\"statistic\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0002","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 13/5, 56/25, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 13/5, 56/25, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064002,"parameters":{"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"parameter\":1,\"statistic\":[0,1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0003","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 11/3, 29/9, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 11/3, 29/9, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064003,"parameters":{"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"parameter\":0,\"statistic\":[0,1,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0004","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 71/17, 1606/289, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 71/17, 1606/289, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064004,"parameters":{"laws":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],"statistic":[0,1,5,6],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"parameter\":1,\"statistic\":[0,1,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0005","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 38/7, 313/49, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 38/7, 313/49, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064005,"parameters":{"laws":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"parameter\":0,\"statistic\":[0,1,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0006","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 111/19, 3620/361, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 111/19, 3620/361, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064006,"parameters":{"laws":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"parameter\":1,\"statistic\":[0,1,7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0007","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 29/4, 163/16, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 29/4, 163/16, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064007,"parameters":{"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"parameter\":0,\"statistic\":[0,1,8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0008","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 53/7, 754/49, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 53/7, 754/49, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064008,"parameters":{"laws":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"parameter\":1,\"statistic\":[0,1,9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0009","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 82/9, 1169/81, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 82/9, 1169/81, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064009,"parameters":{"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"parameter\":0,\"statistic\":[0,1,10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0010","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 215/23, 11344/529, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 215/23, 11344/529, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064010,"parameters":{"laws":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"parameter\":1,\"statistic\":[0,1,11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0011","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 11, 19, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 11, 19, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064011,"parameters":{"laws":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],"statistic":[0,1,12,13],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"parameter\":0,\"statistic\":[0,1,12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0012","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 279/25, 17534/625, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 279/25, 17534/625, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064012,"parameters":{"laws":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"parameter\":1,\"statistic\":[0,1,13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0013","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 142/11, 2881/121, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 142/11, 2881/121, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064013,"parameters":{"laws":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"parameter\":0,\"statistic\":[0,1,14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0014","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 13, 316/9, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 13, 316/9, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064014,"parameters":{"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"parameter\":1,\"statistic\":[0,1,15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0015","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 89/6, 1037/36, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 89/6, 1037/36, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064015,"parameters":{"laws":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"parameter\":0,\"statistic\":[0,1,16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0016","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 431/29, 35770/841, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 431/29, 35770/841, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064016,"parameters":{"laws":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"parameter\":1,\"statistic\":[0,1,17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0017","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 218/13, 5737/169, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 218/13, 5737/169, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064017,"parameters":{"laws":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"parameter\":0,\"statistic\":[0,1,18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0018","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 519/31, 48296/961, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 519/31, 48296/961, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064018,"parameters":{"laws":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"parameter\":1,\"statistic\":[0,1,19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0019","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 131/7, 1921/49, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 131/7, 1921/49, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064019,"parameters":{"laws":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"parameter\":0,\"statistic\":[0,1,20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0020","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 205/11, 7046/121, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 205/11, 7046/121, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064020,"parameters":{"laws":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"parameter\":1,\"statistic\":[0,1,21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0021","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 62/3, 401/9, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 62/3, 401/9, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064021,"parameters":{"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"parameter\":0,\"statistic\":[0,1,22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0022","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 719/35, 81364/1225, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 719/35, 81364/1225, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064022,"parameters":{"laws":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],"statistic":[0,1,23,24],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"parameter\":1,\"statistic\":[0,1,23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0023","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 181/8, 3199/64, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 181/8, 3199/64, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064023,"parameters":{"laws":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"parameter\":0,\"statistic\":[0,1,24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0024","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 831/37, 102386/1369, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(1, 831/37, 102386/1369, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064024,"parameters":{"laws":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"parameter\":1,\"statistic\":[0,1,25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0025","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-at-parameter；資料={\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 418/17, 16033/289, 是, same_theta_sampling_law_support_and_mean)","explanation_zh":"Wave5封閉層核對=(0, 418/17, 16033/289, 是, same_theta_sampling_law_support_and_mean)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-at-parameter","seed":202611064025,"parameters":{"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-at-parameter","semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"parameter\":0,\"statistic\":[0,1,26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0026","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2, 1], [1, 13/7, 55/49]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 2, 1], [1, 13/7, 55/49]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064026,"parameters":{"laws":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],"statistic":[0,1,2,3],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0027","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064027,"parameters":{"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0028","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 11/3, 29/9], [1, 27/8, 239/64]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 11/3, 29/9], [1, 27/8, 239/64]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064028,"parameters":{"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0029","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 59/13, 796/169], [1, 71/17, 1606/289]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 59/13, 796/169], [1, 71/17, 1606/289]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064029,"parameters":{"laws":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],"statistic":[0,1,5,6],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0030","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064030,"parameters":{"laws":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0031","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 19/3, 74/9], [1, 111/19, 3620/361]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 19/3, 74/9], [1, 111/19, 3620/361]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064031,"parameters":{"laws":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0032","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 29/4, 163/16], [1, 67/10, 1261/100]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 29/4, 163/16], [1, 67/10, 1261/100]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064032,"parameters":{"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0033","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 139/17, 3544/289], [1, 53/7, 754/49]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 139/17, 3544/289], [1, 53/7, 754/49]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064033,"parameters":{"laws":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0034","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 82/9, 1169/81], [1, 93/11, 2219/121]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 82/9, 1169/81], [1, 93/11, 2219/121]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064034,"parameters":{"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0035","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 191/19, 6022/361], [1, 215/23, 11344/529]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 191/19, 6022/361], [1, 215/23, 11344/529]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064035,"parameters":{"laws":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0036","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 11, 19], [1, 41/4, 395/16]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 11, 19], [1, 41/4, 395/16]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064036,"parameters":{"laws":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],"statistic":[0,1,12,13],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0037","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 251/21, 9428/441], [1, 279/25, 17534/625]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 251/21, 9428/441], [1, 279/25, 17534/625]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064037,"parameters":{"laws":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0038","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 142/11, 2881/121], [1, 157/13, 5329/169]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 142/11, 2881/121], [1, 157/13, 5329/169]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064038,"parameters":{"laws":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0039","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064039,"parameters":{"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0040","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 89/6, 1037/36], [1, 195/14, 7601/196]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 89/6, 1037/36], [1, 195/14, 7601/196]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064040,"parameters":{"laws":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0041","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 79/5, 784/25], [1, 431/29, 35770/841]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 79/5, 784/25], [1, 431/29, 35770/841]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064041,"parameters":{"laws":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0042","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 218/13, 5737/169], [1, 79/5, 1159/25]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 218/13, 5737/169], [1, 79/5, 1159/25]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064042,"parameters":{"laws":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0043","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 479/27, 26654/729], [1, 519/31, 48296/961]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 479/27, 26654/729], [1, 519/31, 48296/961]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064043,"parameters":{"laws":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0044","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 131/7, 1921/49], [1, 283/16, 13879/256]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 131/7, 1921/49], [1, 283/16, 13879/256]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064044,"parameters":{"laws":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0045","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 571/29, 35212/841], [1, 205/11, 7046/121]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 571/29, 35212/841], [1, 205/11, 7046/121]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064045,"parameters":{"laws":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0046","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 62/3, 401/9], [1, 333/17, 18005/289]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 62/3, 401/9], [1, 333/17, 18005/289]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064046,"parameters":{"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0047","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 671/31, 45418/961], [1, 719/35, 81364/1225]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 671/31, 45418/961], [1, 719/35, 81364/1225]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064047,"parameters":{"laws":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],"statistic":[0,1,23,24],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0048","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 181/8, 3199/64], [1, 43/2, 847/12]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 181/8, 3199/64], [1, 43/2, 847/12]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064048,"parameters":{"laws":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0049","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064049,"parameters":{"laws":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0050","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-family-readback；資料={\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 418/17, 16033/289], [1, 445/19, 28531/361]], variance_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 418/17, 16033/289], [1, 445/19, 28531/361]], variance_is_parameter_indexed_family)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-family-readback","seed":202611064050,"parameters":{"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"domain":"Finite_parametric_estimator_variance","operation":"estvar-family-readback","semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0051","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 5, 2, 1, 1, 是)","explanation_zh":"Wave5封閉層核對=(0, 5, 2, 1, 1, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064051,"parameters":{"laws":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],"statistic":[0,1,2,3],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"parameter\":0,\"statistic\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0052","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 9, 13/5, 56/25, 56/25, 是)","explanation_zh":"Wave5封閉層核對=(1, 9, 13/5, 56/25, 56/25, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064052,"parameters":{"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"parameter\":1,\"statistic\":[0,1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0053","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 50/3, 11/3, 29/9, 29/9, 是)","explanation_zh":"Wave5封閉層核對=(0, 50/3, 11/3, 29/9, 29/9, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064053,"parameters":{"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"parameter\":0,\"statistic\":[0,1,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0054","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 23, 71/17, 1606/289, 1606/289, 是)","explanation_zh":"Wave5封閉層核對=(1, 23, 71/17, 1606/289, 1606/289, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064054,"parameters":{"laws":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],"statistic":[0,1,5,6],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"parameter\":1,\"statistic\":[0,1,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0055","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 251/7, 38/7, 313/49, 313/49, 是)","explanation_zh":"Wave5封閉層核對=(0, 251/7, 38/7, 313/49, 313/49, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064055,"parameters":{"laws":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"parameter\":0,\"statistic\":[0,1,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0056","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 839/19, 111/19, 3620/361, 3620/361, 是)","explanation_zh":"Wave5封閉層核對=(1, 839/19, 111/19, 3620/361, 3620/361, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064056,"parameters":{"laws":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"parameter\":1,\"statistic\":[0,1,7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0057","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 251/4, 29/4, 163/16, 163/16, 是)","explanation_zh":"Wave5封閉層核對=(0, 251/4, 29/4, 163/16, 163/16, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064057,"parameters":{"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"parameter\":0,\"statistic\":[0,1,8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0058","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 509/7, 53/7, 754/49, 754/49, 是)","explanation_zh":"Wave5封閉層核對=(1, 509/7, 53/7, 754/49, 754/49, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064058,"parameters":{"laws":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"parameter\":1,\"statistic\":[0,1,9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0059","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 877/9, 82/9, 1169/81, 1169/81, 是)","explanation_zh":"Wave5封閉層核對=(0, 877/9, 82/9, 1169/81, 1169/81, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064059,"parameters":{"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"parameter\":0,\"statistic\":[0,1,10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0060","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 2503/23, 215/23, 11344/529, 11344/529, 是)","explanation_zh":"Wave5封閉層核對=(1, 2503/23, 215/23, 11344/529, 11344/529, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064060,"parameters":{"laws":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"parameter\":1,\"statistic\":[0,1,11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0061","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 140, 11, 19, 19, 是)","explanation_zh":"Wave5封閉層核對=(0, 140, 11, 19, 19, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064061,"parameters":{"laws":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],"statistic":[0,1,12,13],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"parameter\":0,\"statistic\":[0,1,12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0062","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 763/5, 279/25, 17534/625, 17534/625, 是)","explanation_zh":"Wave5封閉層核對=(1, 763/5, 279/25, 17534/625, 17534/625, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064062,"parameters":{"laws":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"parameter\":1,\"statistic\":[0,1,13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0063","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2095/11, 142/11, 2881/121, 2881/121, 是)","explanation_zh":"Wave5封閉層核對=(0, 2095/11, 142/11, 2881/121, 2881/121, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064063,"parameters":{"laws":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"parameter\":0,\"statistic\":[0,1,14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0064","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1837/9, 13, 316/9, 316/9, 是)","explanation_zh":"Wave5封閉層核對=(1, 1837/9, 13, 316/9, 316/9, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064064,"parameters":{"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"parameter\":1,\"statistic\":[0,1,15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0065","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1493/6, 89/6, 1037/36, 1037/36, 是)","explanation_zh":"Wave5封閉層核對=(0, 1493/6, 89/6, 1037/36, 1037/36, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064065,"parameters":{"laws":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"parameter\":0,\"statistic\":[0,1,16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0066","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 7639/29, 431/29, 35770/841, 35770/841, 是)","explanation_zh":"Wave5封閉層核對=(1, 7639/29, 431/29, 35770/841, 35770/841, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064066,"parameters":{"laws":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"parameter\":1,\"statistic\":[0,1,17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0067","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4097/13, 218/13, 5737/169, 5737/169, 是)","explanation_zh":"Wave5封閉層核對=(0, 4097/13, 218/13, 5737/169, 5737/169, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064067,"parameters":{"laws":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"parameter\":0,\"statistic\":[0,1,18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0068","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 10247/31, 519/31, 48296/961, 48296/961, 是)","explanation_zh":"Wave5封閉層核對=(1, 10247/31, 519/31, 48296/961, 48296/961, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064068,"parameters":{"laws":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"parameter\":1,\"statistic\":[0,1,19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0069","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2726/7, 131/7, 1921/49, 1921/49, 是)","explanation_zh":"Wave5封閉層核對=(0, 2726/7, 131/7, 1921/49, 1921/49, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064069,"parameters":{"laws":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"parameter\":0,\"statistic\":[0,1,20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0070","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 4461/11, 205/11, 7046/121, 7046/121, 是)","explanation_zh":"Wave5封閉層核對=(1, 4461/11, 205/11, 7046/121, 7046/121, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064070,"parameters":{"laws":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"parameter\":1,\"statistic\":[0,1,21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0071","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 1415/3, 62/3, 401/9, 401/9, 是)","explanation_zh":"Wave5封閉層核對=(0, 1415/3, 62/3, 401/9, 401/9, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064071,"parameters":{"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"parameter\":0,\"statistic\":[0,1,22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0072","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3419/7, 719/35, 81364/1225, 81364/1225, 是)","explanation_zh":"Wave5封閉層核對=(1, 3419/7, 719/35, 81364/1225, 81364/1225, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064072,"parameters":{"laws":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],"statistic":[0,1,23,24],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"parameter\":1,\"statistic\":[0,1,23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0073","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 4495/8, 181/8, 3199/64, 3199/64, 是)","explanation_zh":"Wave5封閉層核對=(0, 4495/8, 181/8, 3199/64, 3199/64, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064073,"parameters":{"laws":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"parameter\":0,\"statistic\":[0,1,24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0074","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"parameter\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 21431/37, 831/37, 102386/1369, 102386/1369, 是)","explanation_zh":"Wave5封閉層核對=(1, 21431/37, 831/37, 102386/1369, 102386/1369, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064074,"parameters":{"laws":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"parameter":1,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"parameter\":1,\"statistic\":[0,1,25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0075","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-second-moment-identity；資料={\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"parameter\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 11221/17, 418/17, 16033/289, 16033/289, 是)","explanation_zh":"Wave5封閉層核對=(0, 11221/17, 418/17, 16033/289, 16033/289, 是)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064075,"parameters":{"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"parameter":0,"domain":"Finite_parametric_estimator_variance","operation":"estvar-second-moment-identity","semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"parameter\":0,\"statistic\":[0,1,26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0076","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1], [1, 55/49]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 1], [1, 55/49]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064076,"parameters":{"laws":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],"statistic":[0,1,2,3],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0077","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 238/121], [1, 56/25]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 238/121], [1, 56/25]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064077,"parameters":{"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0078","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 29/9], [1, 239/64]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 29/9], [1, 239/64]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064078,"parameters":{"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0079","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 796/169], [1, 1606/289]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 796/169], [1, 1606/289]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064079,"parameters":{"laws":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],"statistic":[0,1,5,6],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0080","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 313/49], [1, 23/3]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 313/49], [1, 23/3]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064080,"parameters":{"laws":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0081","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 74/9], [1, 3620/361]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 74/9], [1, 3620/361]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064081,"parameters":{"laws":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0082","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 163/16], [1, 1261/100]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 163/16], [1, 1261/100]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064082,"parameters":{"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0083","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3544/289], [1, 754/49]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 3544/289], [1, 754/49]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064083,"parameters":{"laws":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0084","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1169/81], [1, 2219/121]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 1169/81], [1, 2219/121]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064084,"parameters":{"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0085","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 6022/361], [1, 11344/529]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 6022/361], [1, 11344/529]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064085,"parameters":{"laws":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0086","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 19], [1, 395/16]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 19], [1, 395/16]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064086,"parameters":{"laws":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],"statistic":[0,1,12,13],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0087","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 9428/441], [1, 17534/625]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 9428/441], [1, 17534/625]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064087,"parameters":{"laws":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0088","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 2881/121], [1, 5329/169]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 2881/121], [1, 5329/169]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064088,"parameters":{"laws":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0089","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 13906/529], [1, 316/9]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 13906/529], [1, 316/9]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064089,"parameters":{"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0090","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1037/36], [1, 7601/196]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 1037/36], [1, 7601/196]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064090,"parameters":{"laws":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0091","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 784/25], [1, 35770/841]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 784/25], [1, 35770/841]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064091,"parameters":{"laws":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0092","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 5737/169], [1, 1159/25]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 5737/169], [1, 1159/25]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064092,"parameters":{"laws":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0093","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 26654/729], [1, 48296/961]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 26654/729], [1, 48296/961]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064093,"parameters":{"laws":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0094","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 1921/49], [1, 13879/256]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 1921/49], [1, 13879/256]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064094,"parameters":{"laws":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0095","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 35212/841], [1, 7046/121]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 35212/841], [1, 7046/121]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064095,"parameters":{"laws":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0096","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 401/9], [1, 18005/289]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 401/9], [1, 18005/289]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064096,"parameters":{"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0097","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 45418/961], [1, 81364/1225]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 45418/961], [1, 81364/1225]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064097,"parameters":{"laws":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],"statistic":[0,1,23,24],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0098","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 3199/64], [1, 847/12]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 3199/64], [1, 847/12]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064098,"parameters":{"laws":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0099","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 57416/1089], [1, 102386/1369]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 57416/1089], [1, 102386/1369]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064099,"parameters":{"laws":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r04-estimator-variance-0100","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-variance"],"title_zh":"估計量變異數","stem_zh":"共同有限樣本空間與逐theta estimator sampling law。操作=estvar-property-boundary；資料={\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, 16033/289], [1, 28531/361]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation_zh":"Wave5封閉層核對=([[0, 16033/289], [1, 28531/361]], 否, 否, 否, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)；依完整明示資料獨立重算；Var_theta(T)=E_theta[(T-E_theta T)^2]且law support mean綁同一theta；零變異只表示該theta下幾乎必然常數；不推出unbiased consistency或MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量變異數"]},"variant":{"family_id":"qf-mod37-estvar-property-boundary","seed":202611064100,"parameters":{"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"domain":"Finite_parametric_estimator_variance","operation":"estvar-property-boundary","semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、greedy最優性、maximal等於maximum、單一theta無偏、variance推MSE、Fermat converse、有限搜尋完整Diophantine解、唯一quotient代表元、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}