{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261106-r03-estimator-bias-0001","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[1,3]],\"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, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063001,"parameters":{"theta_values":[[1,2],[1,3]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[1,3]]}"}},"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-r03-estimator-bias-0002","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[2,4]],\"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, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063002,"parameters":{"theta_values":[[1,2],[2,4]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[2,4]]}"}},"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-r03-estimator-bias-0003","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[3,5]],\"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, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063003,"parameters":{"theta_values":[[1,2],[3,5]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[3,5]]}"}},"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-r03-estimator-bias-0004","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[4,6]],\"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, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063004,"parameters":{"theta_values":[[1,2],[4,6]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[4,6]]}"}},"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-r03-estimator-bias-0005","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[5,7]],\"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, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063005,"parameters":{"theta_values":[[1,2],[5,7]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[5,7]]}"}},"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-r03-estimator-bias-0006","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[6,8]],\"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, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063006,"parameters":{"theta_values":[[1,2],[6,8]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[6,8]]}"}},"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-r03-estimator-bias-0007","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[7,9]],\"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, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063007,"parameters":{"theta_values":[[1,2],[7,9]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[7,9]]}"}},"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-r03-estimator-bias-0008","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[8,10]],\"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, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063008,"parameters":{"theta_values":[[1,2],[8,10]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[8,10]]}"}},"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-r03-estimator-bias-0009","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[9,11]],\"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, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063009,"parameters":{"theta_values":[[1,2],[9,11]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[9,11]]}"}},"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-r03-estimator-bias-0010","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[10,12]],\"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, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063010,"parameters":{"theta_values":[[1,2],[10,12]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[10,12]]}"}},"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-r03-estimator-bias-0011","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[11,13]],\"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, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063011,"parameters":{"theta_values":[[1,2],[11,13]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[11,13]]}"}},"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-r03-estimator-bias-0012","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[12,14]],\"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, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063012,"parameters":{"theta_values":[[1,2],[12,14]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[12,14]]}"}},"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-r03-estimator-bias-0013","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[13,15]],\"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, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063013,"parameters":{"theta_values":[[1,2],[13,15]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[13,15]]}"}},"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-r03-estimator-bias-0014","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[14,16]],\"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, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063014,"parameters":{"theta_values":[[1,2],[14,16]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[14,16]]}"}},"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-r03-estimator-bias-0015","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[15,17]],\"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, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063015,"parameters":{"theta_values":[[1,2],[15,17]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[15,17]]}"}},"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-r03-estimator-bias-0016","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[16,18]],\"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, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063016,"parameters":{"theta_values":[[1,2],[16,18]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[16,18]]}"}},"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-r03-estimator-bias-0017","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[17,19]],\"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, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063017,"parameters":{"theta_values":[[1,2],[17,19]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[17,19]]}"}},"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-r03-estimator-bias-0018","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[18,20]],\"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, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063018,"parameters":{"theta_values":[[1,2],[18,20]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[18,20]]}"}},"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-r03-estimator-bias-0019","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[19,21]],\"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, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063019,"parameters":{"theta_values":[[1,2],[19,21]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[19,21]]}"}},"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-r03-estimator-bias-0020","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[20,22]],\"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, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063020,"parameters":{"theta_values":[[1,2],[20,22]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[20,22]]}"}},"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-r03-estimator-bias-0021","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[21,23]],\"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, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063021,"parameters":{"theta_values":[[1,2],[21,23]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[21,23]]}"}},"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-r03-estimator-bias-0022","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[22,24]],\"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, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063022,"parameters":{"theta_values":[[1,2],[22,24]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[22,24]]}"}},"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-r03-estimator-bias-0023","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[23,25]],\"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, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063023,"parameters":{"theta_values":[[1,2],[23,25]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[23,25]]}"}},"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-r03-estimator-bias-0024","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[24,26]],\"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, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063024,"parameters":{"theta_values":[[1,2],[24,26]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[24,26]]}"}},"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-r03-estimator-bias-0025","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-at-parameter；資料={\"theta_values\":[[1,2],[25,27]],\"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, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution)","explanation_zh":"Wave5封閉層核對=(0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-at-parameter","seed":202611063025,"parameters":{"theta_values":[[1,2],[25,27]],"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_bias","operation":"bias-at-parameter","semantic_template":"qf-mod37-bias-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],\"theta_values\":[[1,2],[25,27]]}"}},"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-r03-estimator-bias-0026","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[1,3]],\"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, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063026,"parameters":{"theta_values":[[1,2],[1,3]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"}},"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-r03-estimator-bias-0027","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[2,4]],\"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, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063027,"parameters":{"theta_values":[[1,2],[2,4]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"}},"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-r03-estimator-bias-0028","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[3,5]],\"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, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063028,"parameters":{"theta_values":[[1,2],[3,5]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"}},"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-r03-estimator-bias-0029","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[4,6]],\"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, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063029,"parameters":{"theta_values":[[1,2],[4,6]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"}},"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-r03-estimator-bias-0030","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[5,7]],\"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, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063030,"parameters":{"theta_values":[[1,2],[5,7]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"}},"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-r03-estimator-bias-0031","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[6,8]],\"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, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063031,"parameters":{"theta_values":[[1,2],[6,8]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"}},"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-r03-estimator-bias-0032","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[7,9]],\"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, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063032,"parameters":{"theta_values":[[1,2],[7,9]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"}},"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-r03-estimator-bias-0033","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[8,10]],\"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, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063033,"parameters":{"theta_values":[[1,2],[8,10]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"}},"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-r03-estimator-bias-0034","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[9,11]],\"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, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063034,"parameters":{"theta_values":[[1,2],[9,11]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"}},"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-r03-estimator-bias-0035","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[10,12]],\"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, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063035,"parameters":{"theta_values":[[1,2],[10,12]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"}},"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-r03-estimator-bias-0036","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[11,13]],\"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, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063036,"parameters":{"theta_values":[[1,2],[11,13]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"}},"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-r03-estimator-bias-0037","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[12,14]],\"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, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063037,"parameters":{"theta_values":[[1,2],[12,14]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"}},"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-r03-estimator-bias-0038","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[13,15]],\"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, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063038,"parameters":{"theta_values":[[1,2],[13,15]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"}},"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-r03-estimator-bias-0039","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[14,16]],\"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, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063039,"parameters":{"theta_values":[[1,2],[14,16]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"}},"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-r03-estimator-bias-0040","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[15,17]],\"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, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063040,"parameters":{"theta_values":[[1,2],[15,17]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"}},"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-r03-estimator-bias-0041","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[16,18]],\"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, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063041,"parameters":{"theta_values":[[1,2],[16,18]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"}},"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-r03-estimator-bias-0042","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[17,19]],\"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, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063042,"parameters":{"theta_values":[[1,2],[17,19]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"}},"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-r03-estimator-bias-0043","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[18,20]],\"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, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063043,"parameters":{"theta_values":[[1,2],[18,20]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"}},"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-r03-estimator-bias-0044","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[19,21]],\"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, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063044,"parameters":{"theta_values":[[1,2],[19,21]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"}},"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-r03-estimator-bias-0045","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[20,22]],\"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, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063045,"parameters":{"theta_values":[[1,2],[20,22]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"}},"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-r03-estimator-bias-0046","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[21,23]],\"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, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063046,"parameters":{"theta_values":[[1,2],[21,23]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"}},"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-r03-estimator-bias-0047","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[22,24]],\"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, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063047,"parameters":{"theta_values":[[1,2],[22,24]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"}},"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-r03-estimator-bias-0048","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[23,25]],\"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, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063048,"parameters":{"theta_values":[[1,2],[23,25]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"}},"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-r03-estimator-bias-0049","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[24,26]],\"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, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063049,"parameters":{"theta_values":[[1,2],[24,26]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"}},"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-r03-estimator-bias-0050","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-family-readback；資料={\"theta_values\":[[1,2],[25,27]],\"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, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family)","explanation_zh":"Wave5封閉層核對=([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-family-readback","seed":202611063050,"parameters":{"theta_values":[[1,2],[25,27]],"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_bias","operation":"bias-family-readback","semantic_template":"qf-mod37-bias-family-readback/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"}},"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-r03-estimator-bias-0051","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[1,3]],\"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, 3/2], [1, 32/21]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 3/2], [1, 32/21]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063051,"parameters":{"theta_values":[[1,2],[1,3]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"}},"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-r03-estimator-bias-0052","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[2,4]],\"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, 51/22], [1, 21/10]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 51/22], [1, 21/10]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063052,"parameters":{"theta_values":[[1,2],[2,4]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"}},"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-r03-estimator-bias-0053","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[3,5]],\"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, 19/6], [1, 111/40]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 19/6], [1, 111/40]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063053,"parameters":{"theta_values":[[1,2],[3,5]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"}},"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-r03-estimator-bias-0054","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[4,6]],\"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, 105/26], [1, 179/51]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 105/26], [1, 179/51]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063054,"parameters":{"theta_values":[[1,2],[4,6]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"}},"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-r03-estimator-bias-0055","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[5,7]],\"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, 69/14], [1, 30/7]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 69/14], [1, 30/7]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063055,"parameters":{"theta_values":[[1,2],[5,7]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"}},"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-r03-estimator-bias-0056","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[6,8]],\"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, 35/6], [1, 387/76]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 35/6], [1, 387/76]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063056,"parameters":{"theta_values":[[1,2],[6,8]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"}},"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-r03-estimator-bias-0057","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[7,9]],\"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, 27/4], [1, 533/90]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 27/4], [1, 533/90]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063057,"parameters":{"theta_values":[[1,2],[7,9]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"}},"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-r03-estimator-bias-0058","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[8,10]],\"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, 261/34], [1, 237/35]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 261/34], [1, 237/35]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063058,"parameters":{"theta_values":[[1,2],[8,10]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"}},"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-r03-estimator-bias-0059","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[9,11]],\"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, 155/18], [1, 84/11]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 155/18], [1, 84/11]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063059,"parameters":{"theta_values":[[1,2],[9,11]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"}},"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-r03-estimator-bias-0060","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[10,12]],\"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, 363/38], [1, 1175/138]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 363/38], [1, 1175/138]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063060,"parameters":{"theta_values":[[1,2],[10,12]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"}},"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-r03-estimator-bias-0061","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[11,13]],\"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, 21/2], [1, 489/52]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 21/2], [1, 489/52]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063061,"parameters":{"theta_values":[[1,2],[11,13]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"}},"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-r03-estimator-bias-0062","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[12,14]],\"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, 481/42], [1, 1803/175]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 481/42], [1, 1803/175]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063062,"parameters":{"theta_values":[[1,2],[12,14]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"}},"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-r03-estimator-bias-0063","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[13,15]],\"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, 273/22], [1, 2186/195]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 273/22], [1, 2186/195]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063063,"parameters":{"theta_values":[[1,2],[13,15]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"}},"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-r03-estimator-bias-0064","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[14,16]],\"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, 615/46], [1, 97/8]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 615/46], [1, 97/8]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063064,"parameters":{"theta_values":[[1,2],[14,16]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"}},"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-r03-estimator-bias-0065","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[15,17]],\"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, 43/3], [1, 3105/238]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 43/3], [1, 3105/238]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063065,"parameters":{"theta_values":[[1,2],[15,17]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"}},"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-r03-estimator-bias-0066","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[16,18]],\"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, 153/10], [1, 3647/261]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 153/10], [1, 3647/261]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063066,"parameters":{"theta_values":[[1,2],[16,18]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"}},"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-r03-estimator-bias-0067","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[17,19]],\"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, 423/26], [1, 1416/95]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 423/26], [1, 1416/95]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063067,"parameters":{"theta_values":[[1,2],[17,19]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"}},"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-r03-estimator-bias-0068","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[18,20]],\"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, 931/54], [1, 4911/310]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 931/54], [1, 4911/310]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063068,"parameters":{"theta_values":[[1,2],[18,20]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"}},"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-r03-estimator-bias-0069","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[19,21]],\"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, 255/14], [1, 5639/336]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 255/14], [1, 5639/336]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063069,"parameters":{"theta_values":[[1,2],[19,21]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"}},"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-r03-estimator-bias-0070","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[20,22]],\"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, 1113/58], [1, 195/11]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 1113/58], [1, 195/11]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063070,"parameters":{"theta_values":[[1,2],[20,22]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"}},"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-r03-estimator-bias-0071","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[21,23]],\"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, 121/6], [1, 7302/391]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 121/6], [1, 7302/391]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063071,"parameters":{"theta_values":[[1,2],[21,23]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"}},"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-r03-estimator-bias-0072","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[22,24]],\"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, 1311/62], [1, 8243/420]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 1311/62], [1, 8243/420]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063072,"parameters":{"theta_values":[[1,2],[22,24]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"}},"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-r03-estimator-bias-0073","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[23,25]],\"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, 177/8], [1, 1029/50]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 177/8], [1, 1029/50]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063073,"parameters":{"theta_values":[[1,2],[23,25]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"}},"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-r03-estimator-bias-0074","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[24,26]],\"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, 1525/66], [1, 10359/481]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 1525/66], [1, 10359/481]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063074,"parameters":{"theta_values":[[1,2],[24,26]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"}},"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-r03-estimator-bias-0075","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-unbiased-audit；資料={\"theta_values\":[[1,2],[25,27]],\"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, 819/34], [1, 11540/513]], 否, unbiased_requires_every_theta)","explanation_zh":"Wave5封閉層核對=([[0, 819/34], [1, 11540/513]], 否, unbiased_requires_every_theta)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-unbiased-audit","seed":202611063075,"parameters":{"theta_values":[[1,2],[25,27]],"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_bias","operation":"bias-unbiased-audit","semantic_template":"qf-mod37-bias-unbiased-audit/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"}},"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-r03-estimator-bias-0076","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[1,3]],\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 3/2, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 3/2, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063076,"parameters":{"theta_values":[[1,2],[1,3]],"laws":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],"statistic":[0,1,2,3],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,2,3],\"theta_values\":[[1,2],[1,3]]}"}},"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-r03-estimator-bias-0077","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[2,4]],\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4],\"parameter\":1,\"outcome\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 21/10, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 1, 1, 21/10, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063077,"parameters":{"theta_values":[[1,2],[2,4]],"laws":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],"statistic":[0,1,3,4],"parameter":1,"outcome":1,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,3,4],\"theta_values\":[[1,2],[2,4]]}"}},"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-r03-estimator-bias-0078","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[3,5]],\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5],\"parameter\":0,\"outcome\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 4, 19/6, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 2, 4, 19/6, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063078,"parameters":{"theta_values":[[1,2],[3,5]],"laws":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],"statistic":[0,1,4,5],"parameter":0,"outcome":2,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,4,5],\"theta_values\":[[1,2],[3,5]]}"}},"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-r03-estimator-bias-0079","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[4,6]],\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6],\"parameter\":1,\"outcome\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 6, 179/51, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 3, 6, 179/51, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063079,"parameters":{"theta_values":[[1,2],[4,6]],"laws":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],"statistic":[0,1,5,6],"parameter":1,"outcome":3,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,5,6],\"theta_values\":[[1,2],[4,6]]}"}},"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-r03-estimator-bias-0080","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[5,7]],\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 69/14, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 69/14, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063080,"parameters":{"theta_values":[[1,2],[5,7]],"laws":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],"statistic":[0,1,6,7],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,6,7],\"theta_values\":[[1,2],[5,7]]}"}},"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-r03-estimator-bias-0081","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[6,8]],\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8],\"parameter\":1,\"outcome\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 387/76, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 1, 1, 387/76, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063081,"parameters":{"theta_values":[[1,2],[6,8]],"laws":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],"statistic":[0,1,7,8],"parameter":1,"outcome":1,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,7,8],\"theta_values\":[[1,2],[6,8]]}"}},"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-r03-estimator-bias-0082","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[7,9]],\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9],\"parameter\":0,\"outcome\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 8, 27/4, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 2, 8, 27/4, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063082,"parameters":{"theta_values":[[1,2],[7,9]],"laws":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],"statistic":[0,1,8,9],"parameter":0,"outcome":2,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,8,9],\"theta_values\":[[1,2],[7,9]]}"}},"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-r03-estimator-bias-0083","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[8,10]],\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10],\"parameter\":1,\"outcome\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 10, 237/35, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 3, 10, 237/35, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063083,"parameters":{"theta_values":[[1,2],[8,10]],"laws":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],"statistic":[0,1,9,10],"parameter":1,"outcome":3,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,9,10],\"theta_values\":[[1,2],[8,10]]}"}},"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-r03-estimator-bias-0084","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[9,11]],\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 155/18, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 155/18, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063084,"parameters":{"theta_values":[[1,2],[9,11]],"laws":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],"statistic":[0,1,10,11],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,10,11],\"theta_values\":[[1,2],[9,11]]}"}},"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-r03-estimator-bias-0085","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[10,12]],\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12],\"parameter\":1,\"outcome\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 1175/138, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 1, 1, 1175/138, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063085,"parameters":{"theta_values":[[1,2],[10,12]],"laws":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],"statistic":[0,1,11,12],"parameter":1,"outcome":1,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,11,12],\"theta_values\":[[1,2],[10,12]]}"}},"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-r03-estimator-bias-0086","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[11,13]],\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13],\"parameter\":0,\"outcome\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 12, 21/2, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 2, 12, 21/2, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063086,"parameters":{"theta_values":[[1,2],[11,13]],"laws":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],"statistic":[0,1,12,13],"parameter":0,"outcome":2,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,12,13],\"theta_values\":[[1,2],[11,13]]}"}},"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-r03-estimator-bias-0087","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[12,14]],\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14],\"parameter\":1,\"outcome\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 14, 1803/175, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 3, 14, 1803/175, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063087,"parameters":{"theta_values":[[1,2],[12,14]],"laws":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],"statistic":[0,1,13,14],"parameter":1,"outcome":3,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,13,14],\"theta_values\":[[1,2],[12,14]]}"}},"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-r03-estimator-bias-0088","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[13,15]],\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 273/22, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 273/22, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063088,"parameters":{"theta_values":[[1,2],[13,15]],"laws":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],"statistic":[0,1,14,15],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,14,15],\"theta_values\":[[1,2],[13,15]]}"}},"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-r03-estimator-bias-0089","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[14,16]],\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16],\"parameter\":1,\"outcome\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 97/8, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 1, 1, 97/8, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063089,"parameters":{"theta_values":[[1,2],[14,16]],"laws":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],"statistic":[0,1,15,16],"parameter":1,"outcome":1,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,15,16],\"theta_values\":[[1,2],[14,16]]}"}},"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-r03-estimator-bias-0090","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[15,17]],\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17],\"parameter\":0,\"outcome\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 16, 43/3, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 2, 16, 43/3, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063090,"parameters":{"theta_values":[[1,2],[15,17]],"laws":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],"statistic":[0,1,16,17],"parameter":0,"outcome":2,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,16,17],\"theta_values\":[[1,2],[15,17]]}"}},"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-r03-estimator-bias-0091","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[16,18]],\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18],\"parameter\":1,\"outcome\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 18, 3647/261, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 3, 18, 3647/261, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063091,"parameters":{"theta_values":[[1,2],[16,18]],"laws":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],"statistic":[0,1,17,18],"parameter":1,"outcome":3,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,17,18],\"theta_values\":[[1,2],[16,18]]}"}},"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-r03-estimator-bias-0092","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[17,19]],\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 423/26, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 423/26, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063092,"parameters":{"theta_values":[[1,2],[17,19]],"laws":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],"statistic":[0,1,18,19],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,18,19],\"theta_values\":[[1,2],[17,19]]}"}},"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-r03-estimator-bias-0093","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[18,20]],\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20],\"parameter\":1,\"outcome\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 4911/310, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 1, 1, 4911/310, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063093,"parameters":{"theta_values":[[1,2],[18,20]],"laws":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],"statistic":[0,1,19,20],"parameter":1,"outcome":1,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,19,20],\"theta_values\":[[1,2],[18,20]]}"}},"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-r03-estimator-bias-0094","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[19,21]],\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21],\"parameter\":0,\"outcome\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 20, 255/14, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 2, 20, 255/14, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063094,"parameters":{"theta_values":[[1,2],[19,21]],"laws":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],"statistic":[0,1,20,21],"parameter":0,"outcome":2,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,20,21],\"theta_values\":[[1,2],[19,21]]}"}},"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-r03-estimator-bias-0095","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[20,22]],\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22],\"parameter\":1,\"outcome\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 22, 195/11, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 3, 22, 195/11, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063095,"parameters":{"theta_values":[[1,2],[20,22]],"laws":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],"statistic":[0,1,21,22],"parameter":1,"outcome":3,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,21,22],\"theta_values\":[[1,2],[20,22]]}"}},"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-r03-estimator-bias-0096","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[21,23]],\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 121/6, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 121/6, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063096,"parameters":{"theta_values":[[1,2],[21,23]],"laws":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],"statistic":[0,1,22,23],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,22,23],\"theta_values\":[[1,2],[21,23]]}"}},"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-r03-estimator-bias-0097","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[22,24]],\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24],\"parameter\":1,\"outcome\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 1, 1, 8243/420, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 1, 1, 8243/420, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063097,"parameters":{"theta_values":[[1,2],[22,24]],"laws":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],"statistic":[0,1,23,24],"parameter":1,"outcome":1,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"outcome\":1,\"parameter\":1,\"statistic\":[0,1,23,24],\"theta_values\":[[1,2],[22,24]]}"}},"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-r03-estimator-bias-0098","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[23,25]],\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25],\"parameter\":0,\"outcome\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 2, 24, 177/8, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 2, 24, 177/8, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063098,"parameters":{"theta_values":[[1,2],[23,25]],"laws":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],"statistic":[0,1,24,25],"parameter":0,"outcome":2,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"outcome\":2,\"parameter\":0,\"statistic\":[0,1,24,25],\"theta_values\":[[1,2],[23,25]]}"}},"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-r03-estimator-bias-0099","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[24,26]],\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26],\"parameter\":1,\"outcome\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 3, 26, 10359/481, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(1, 3, 26, 10359/481, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063099,"parameters":{"theta_values":[[1,2],[24,26]],"laws":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],"statistic":[0,1,25,26],"parameter":1,"outcome":3,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"outcome\":3,\"parameter\":1,\"statistic\":[0,1,25,26],\"theta_values\":[[1,2],[24,26]]}"}},"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-r03-estimator-bias-0100","batch_id":"batch-20261014-advance-20261106-r03-estimator-bias-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-bias"],"title_zh":"估計量偏誤","stem_zh":"共同有限樣本空間、逐theta sampling law與同一total estimator。操作=bias-observed-boundary；資料={\"theta_values\":[[1,2],[25,27]],\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27],\"parameter\":0,\"outcome\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, 0, 0, 819/34, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)","explanation_zh":"Wave5封閉層核對=(0, 0, 0, 819/34, 否, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)；依完整明示資料獨立重算；Bias_theta(T)=E_theta[T]-theta且期望來自同一theta抽樣分布；unbiased要求每個theta；單一estimate不足且不推出零變異一致充分或最小MSE。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量偏誤"]},"variant":{"family_id":"qf-mod37-bias-observed-boundary","seed":202611063100,"parameters":{"theta_values":[[1,2],[25,27]],"laws":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],"statistic":[0,1,26,27],"parameter":0,"outcome":0,"domain":"Finite_parametric_estimator_bias","operation":"bias-observed-boundary","semantic_template":"qf-mod37-bias-observed-boundary/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"outcome\":0,\"parameter\":0,\"statistic\":[0,1,26,27],\"theta_values\":[[1,2],[25,27]]}"}},"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"}}