{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0001","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-12,\"support\":[-13,-12,-11],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, [[-13, 1/4, 1, 1/4], [-12, 1/2, 0, 0], [-11, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-12, [[-13, 1/4, 1, 1/4], [-12, 1/2, 0, 0], [-11, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145001,"parameters":{"theta":-12,"support":[-13,-12,-11],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-11,1],[-12,1],[-13,1]],\"theta\":[-12,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0002","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-11,\"support\":[-13,-11,-9],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, [[-13, 1/4, 4, 1], [-11, 1/2, 0, 0], [-9, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-11, [[-13, 1/4, 4, 1], [-11, 1/2, 0, 0], [-9, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145002,"parameters":{"theta":-11,"support":[-13,-11,-9],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-11,1],[-13,1],[-9,1]],\"theta\":[-11,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0003","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-10,\"support\":[-13,-10,-7],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145003,"parameters":{"theta":-10,"support":[-13,-10,-7],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0004","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-9,\"support\":[-13,-9,-5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, [[-13, 1/4, 16, 4], [-9, 1/2, 0, 0], [-5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-9, [[-13, 1/4, 16, 4], [-9, 1/2, 0, 0], [-5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145004,"parameters":{"theta":-9,"support":[-13,-9,-5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-5,1],[-9,1]],\"theta\":[-9,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0005","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-8,\"support\":[-13,-8,-3],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, [[-13, 1/4, 25, 25/4], [-8, 1/2, 0, 0], [-3, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-8, [[-13, 1/4, 25, 25/4], [-8, 1/2, 0, 0], [-3, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145005,"parameters":{"theta":-8,"support":[-13,-8,-3],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-3,1],[-8,1]],\"theta\":[-8,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0006","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-7,\"support\":[-8,-7,-6],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, [[-8, 1/4, 1, 1/4], [-7, 1/2, 0, 0], [-6, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-7, [[-8, 1/4, 1, 1/4], [-7, 1/2, 0, 0], [-6, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145006,"parameters":{"theta":-7,"support":[-8,-7,-6],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-6,1],[-7,1],[-8,1]],\"theta\":[-7,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0007","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-6,\"support\":[-8,-6,-4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145007,"parameters":{"theta":-6,"support":[-8,-6,-4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-6,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0008","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-5,\"support\":[-8,-5,-2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, [[-8, 1/4, 9, 9/4], [-5, 1/2, 0, 0], [-2, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-5, [[-8, 1/4, 9, 9/4], [-5, 1/2, 0, 0], [-2, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145008,"parameters":{"theta":-5,"support":[-8,-5,-2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-2,1],[-5,1],[-8,1]],\"theta\":[-5,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0009","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-4,\"support\":[-8,-4,0],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, [[-8, 1/4, 16, 4], [-4, 1/2, 0, 0], [0, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-4, [[-8, 1/4, 16, 4], [-4, 1/2, 0, 0], [0, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145009,"parameters":{"theta":-4,"support":[-8,-4,0],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-4,1],[-8,1],[0,1]],\"theta\":[-4,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0010","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-3,\"support\":[-8,-3,2],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, [[-8, 1/4, 25, 25/4], [-3, 1/2, 0, 0], [2, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-3, [[-8, 1/4, 25, 25/4], [-3, 1/2, 0, 0], [2, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145010,"parameters":{"theta":-3,"support":[-8,-3,2],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-3,1],[-8,1],[2,1]],\"theta\":[-3,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0011","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-2,\"support\":[-3,-2,-1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145011,"parameters":{"theta":-2,"support":[-3,-2,-1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-2,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0012","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":-1,\"support\":[-3,-1,1],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, [[-3, 1/4, 4, 1], [-1, 1/2, 0, 0], [1, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(-1, [[-3, 1/4, 4, 1], [-1, 1/2, 0, 0], [1, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145012,"parameters":{"theta":-1,"support":[-3,-1,1],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-1,1],[-3,1],[1,1]],\"theta\":[-1,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0013","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":0,\"support\":[-3,0,3],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145013,"parameters":{"theta":0,"support":[-3,0,3],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[0,1],[3,1]],\"theta\":[0,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0014","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":1,\"support\":[-3,1,5],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[-3, 1/4, 16, 4], [1, 1/2, 0, 0], [5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(1, [[-3, 1/4, 16, 4], [1, 1/2, 0, 0], [5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145014,"parameters":{"theta":1,"support":[-3,1,5],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[1,1],[5,1]],\"theta\":[1,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0015","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":2,\"support\":[-3,2,7],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[-3, 1/4, 25, 25/4], [2, 1/2, 0, 0], [7, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(2, [[-3, 1/4, 25, 25/4], [2, 1/2, 0, 0], [7, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145015,"parameters":{"theta":2,"support":[-3,2,7],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[2,1],[7,1]],\"theta\":[2,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0016","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":3,\"support\":[2,3,4],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145016,"parameters":{"theta":3,"support":[2,3,4],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[3,1],[4,1]],\"theta\":[3,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0017","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":4,\"support\":[2,4,6],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[2, 1/4, 4, 1], [4, 1/2, 0, 0], [6, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(4, [[2, 1/4, 4, 1], [4, 1/2, 0, 0], [6, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145017,"parameters":{"theta":4,"support":[2,4,6],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[4,1],[6,1]],\"theta\":[4,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0018","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":5,\"support\":[2,5,8],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[2, 1/4, 9, 9/4], [5, 1/2, 0, 0], [8, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(5, [[2, 1/4, 9, 9/4], [5, 1/2, 0, 0], [8, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145018,"parameters":{"theta":5,"support":[2,5,8],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[5,1],[8,1]],\"theta\":[5,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0019","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":6,\"support\":[2,6,10],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[2, 1/4, 16, 4], [6, 1/2, 0, 0], [10, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(6, [[2, 1/4, 16, 4], [6, 1/2, 0, 0], [10, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145019,"parameters":{"theta":6,"support":[2,6,10],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[10,1],[2,1],[6,1]],\"theta\":[6,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0020","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":7,\"support\":[2,7,12],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[2, 1/4, 25, 25/4], [7, 1/2, 0, 0], [12, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(7, [[2, 1/4, 25, 25/4], [7, 1/2, 0, 0], [12, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145020,"parameters":{"theta":7,"support":[2,7,12],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[12,1],[2,1],[7,1]],\"theta\":[7,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0021","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":8,\"support\":[7,8,9],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[7, 1/4, 1, 1/4], [8, 1/2, 0, 0], [9, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(8, [[7, 1/4, 1, 1/4], [8, 1/2, 0, 0], [9, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145021,"parameters":{"theta":8,"support":[7,8,9],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[7,1],[8,1],[9,1]],\"theta\":[8,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0022","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":9,\"support\":[7,9,11],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145022,"parameters":{"theta":9,"support":[7,9,11],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[7,1],[9,1]],\"theta\":[9,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0023","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":10,\"support\":[7,10,13],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[7, 1/4, 9, 9/4], [10, 1/2, 0, 0], [13, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(10, [[7, 1/4, 9, 9/4], [10, 1/2, 0, 0], [13, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145023,"parameters":{"theta":10,"support":[7,10,13],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[13,1],[7,1]],\"theta\":[10,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0024","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":11,\"support\":[7,11,15],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[7, 1/4, 16, 4], [11, 1/2, 0, 0], [15, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(11, [[7, 1/4, 16, 4], [11, 1/2, 0, 0], [15, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145024,"parameters":{"theta":11,"support":[7,11,15],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[11,1],[15,1],[7,1]],\"theta\":[11,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0025","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-single-theta-exact；資料={\"theta\":12,\"support\":[7,12,17],\"probabilities\":[[1,4],[1,2],[1,4]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[7, 1/4, 25, 25/4], [12, 1/2, 0, 0], [17, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation_zh":"Wave6第八層核對=(12, [[7, 1/4, 25, 25/4], [12, 1/2, 0, 0], [17, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-single-theta-exact","seed":202611145025,"parameters":{"theta":12,"support":[7,12,17],"probabilities":[[1,4],[1,2],[1,4]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-single-theta-exact","semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[12,1],[17,1],[7,1]],\"theta\":[12,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0026","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-12,[-13,-12,-11],[[1,4],[1,2],[1,4]]],[-11,[-12,-11,-10],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-11, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], 1/2], [-12, [[-13, 1/4], [-12, 1/2], [-11, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-11, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], 1/2], [-12, [[-13, 1/4], [-12, 1/2], [-11, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145026,"parameters":{"laws":[[-12,[-13,-12,-11],[[1,4],[1,2],[1,4]]],[-11,[-12,-11,-10],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-11,1],[[-10,1],[-11,1],[-12,1]],[[1,4],[1,2],[1,4]]],[[-12,1],[[-11,1],[-12,1],[-13,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0027","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-11,[-13,-11,-9],[[1,4],[1,2],[1,4]]],[-10,[-11,-10,-9],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-10, [[-11, 1/4], [-10, 1/2], [-9, 1/4]], 1/2], [-11, [[-13, 1/4], [-11, 1/2], [-9, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-10, [[-11, 1/4], [-10, 1/2], [-9, 1/4]], 1/2], [-11, [[-13, 1/4], [-11, 1/2], [-9, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145027,"parameters":{"laws":[[-11,[-13,-11,-9],[[1,4],[1,2],[1,4]]],[-10,[-11,-10,-9],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-10,1],[[-10,1],[-11,1],[-9,1]],[[1,2],[1,4],[1,4]]],[[-11,1],[[-11,1],[-13,1],[-9,1]],[[1,2],[1,4],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0028","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-10,[-13,-10,-7],[[1,4],[1,2],[1,4]]],[-9,[-10,-9,-8],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145028,"parameters":{"laws":[[-10,[-13,-10,-7],[[1,4],[1,2],[1,4]]],[-9,[-10,-9,-8],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-10,1],[[-10,1],[-13,1],[-7,1]],[[1,2],[1,4],[1,4]]],[[-9,1],[[-10,1],[-8,1],[-9,1]],[[1,4],[1,4],[1,2]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0029","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-9,[-13,-9,-5],[[1,4],[1,2],[1,4]]],[-8,[-9,-8,-7],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-8, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], 1/2], [-9, [[-13, 1/4], [-9, 1/2], [-5, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-8, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], 1/2], [-9, [[-13, 1/4], [-9, 1/2], [-5, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145029,"parameters":{"laws":[[-9,[-13,-9,-5],[[1,4],[1,2],[1,4]]],[-8,[-9,-8,-7],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-8,1],[[-7,1],[-8,1],[-9,1]],[[1,4],[1,2],[1,4]]],[[-9,1],[[-13,1],[-5,1],[-9,1]],[[1,4],[1,4],[1,2]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0030","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-8,[-13,-8,-3],[[1,4],[1,2],[1,4]]],[-7,[-8,-7,-6],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2], [-8, [[-13, 1/4], [-8, 1/2], [-3, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2], [-8, [[-13, 1/4], [-8, 1/2], [-3, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145030,"parameters":{"laws":[[-8,[-13,-8,-3],[[1,4],[1,2],[1,4]]],[-7,[-8,-7,-6],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-7,1],[[-6,1],[-7,1],[-8,1]],[[1,4],[1,2],[1,4]]],[[-8,1],[[-13,1],[-3,1],[-8,1]],[[1,4],[1,4],[1,2]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0031","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-7,[-8,-7,-6],[[1,4],[1,2],[1,4]]],[-6,[-7,-6,-5],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-6, [[-7, 1/4], [-6, 1/2], [-5, 1/4]], 1/2], [-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-6, [[-7, 1/4], [-6, 1/2], [-5, 1/4]], 1/2], [-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145031,"parameters":{"laws":[[-7,[-8,-7,-6],[[1,4],[1,2],[1,4]]],[-6,[-7,-6,-5],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-6,1],[[-5,1],[-6,1],[-7,1]],[[1,4],[1,2],[1,4]]],[[-7,1],[[-6,1],[-7,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0032","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-6,[-8,-6,-4],[[1,4],[1,2],[1,4]]],[-5,[-6,-5,-4],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-5, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], 1/2], [-6, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-5, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], 1/2], [-6, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145032,"parameters":{"laws":[[-6,[-8,-6,-4],[[1,4],[1,2],[1,4]]],[-5,[-6,-5,-4],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-5,1],[[-4,1],[-5,1],[-6,1]],[[1,4],[1,2],[1,4]]],[[-6,1],[[-4,1],[-6,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0033","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-5,[-8,-5,-2],[[1,4],[1,2],[1,4]]],[-4,[-5,-4,-3],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-4, [[-5, 1/4], [-4, 1/2], [-3, 1/4]], 1/2], [-5, [[-8, 1/4], [-5, 1/2], [-2, 1/4]], 9/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-4, [[-5, 1/4], [-4, 1/2], [-3, 1/4]], 1/2], [-5, [[-8, 1/4], [-5, 1/2], [-2, 1/4]], 9/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145033,"parameters":{"laws":[[-5,[-8,-5,-2],[[1,4],[1,2],[1,4]]],[-4,[-5,-4,-3],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-4,1],[[-3,1],[-4,1],[-5,1]],[[1,4],[1,2],[1,4]]],[[-5,1],[[-2,1],[-5,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0034","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-4,[-8,-4,0],[[1,4],[1,2],[1,4]]],[-3,[-4,-3,-2],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-3, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1/2], [-4, [[-8, 1/4], [-4, 1/2], [0, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-3, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1/2], [-4, [[-8, 1/4], [-4, 1/2], [0, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145034,"parameters":{"laws":[[-4,[-8,-4,0],[[1,4],[1,2],[1,4]]],[-3,[-4,-3,-2],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-3,1],[[-2,1],[-3,1],[-4,1]],[[1,4],[1,2],[1,4]]],[[-4,1],[[-4,1],[-8,1],[0,1]],[[1,2],[1,4],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0035","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-3,[-8,-3,2],[[1,4],[1,2],[1,4]]],[-2,[-3,-2,-1],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2], [-3, [[-8, 1/4], [-3, 1/2], [2, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2], [-3, [[-8, 1/4], [-3, 1/2], [2, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145035,"parameters":{"laws":[[-3,[-8,-3,2],[[1,4],[1,2],[1,4]]],[-2,[-3,-2,-1],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-2,1],[[-1,1],[-2,1],[-3,1]],[[1,4],[1,2],[1,4]]],[[-3,1],[[-3,1],[-8,1],[2,1]],[[1,2],[1,4],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0036","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-2,[-3,-2,-1],[[1,4],[1,2],[1,4]]],[-1,[-2,-1,0],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1/2], [-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-1, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1/2], [-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145036,"parameters":{"laws":[[-2,[-3,-2,-1],[[1,4],[1,2],[1,4]]],[-1,[-2,-1,0],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-1,1],[[-1,1],[-2,1],[0,1]],[[1,2],[1,4],[1,4]]],[[-2,1],[[-1,1],[-2,1],[-3,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0037","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[-1,[-3,-1,1],[[1,4],[1,2],[1,4]]],[0,[-1,0,1],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[-1, [[-3, 1/4], [-1, 1/2], [1, 1/4]], 2], [0, [[-1, 1/4], [0, 1/2], [1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[-1, [[-3, 1/4], [-1, 1/2], [1, 1/4]], 2], [0, [[-1, 1/4], [0, 1/2], [1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145037,"parameters":{"laws":[[-1,[-3,-1,1],[[1,4],[1,2],[1,4]]],[0,[-1,0,1],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-1,1],[[-1,1],[-3,1],[1,1]],[[1,2],[1,4],[1,4]]],[[0,1],[[-1,1],[0,1],[1,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0038","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[0,[-3,0,3],[[1,4],[1,2],[1,4]]],[1,[0,1,2],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[0, [[-3, 1/4], [0, 1/2], [3, 1/4]], 9/2], [1, [[0, 1/4], [1, 1/2], [2, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[0, [[-3, 1/4], [0, 1/2], [3, 1/4]], 9/2], [1, [[0, 1/4], [1, 1/2], [2, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145038,"parameters":{"laws":[[0,[-3,0,3],[[1,4],[1,2],[1,4]]],[1,[0,1,2],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[0,1],[[-3,1],[0,1],[3,1]],[[1,4],[1,2],[1,4]]],[[1,1],[[0,1],[1,1],[2,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0039","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[1,[-3,1,5],[[1,4],[1,2],[1,4]]],[2,[1,2,3],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[1, [[-3, 1/4], [1, 1/2], [5, 1/4]], 8], [2, [[1, 1/4], [2, 1/2], [3, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[1, [[-3, 1/4], [1, 1/2], [5, 1/4]], 8], [2, [[1, 1/4], [2, 1/2], [3, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145039,"parameters":{"laws":[[1,[-3,1,5],[[1,4],[1,2],[1,4]]],[2,[1,2,3],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[1,1],[[-3,1],[1,1],[5,1]],[[1,4],[1,2],[1,4]]],[[2,1],[[1,1],[2,1],[3,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0040","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[2,[-3,2,7],[[1,4],[1,2],[1,4]]],[3,[2,3,4],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[2, [[-3, 1/4], [2, 1/2], [7, 1/4]], 25/2], [3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[2, [[-3, 1/4], [2, 1/2], [7, 1/4]], 25/2], [3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145040,"parameters":{"laws":[[2,[-3,2,7],[[1,4],[1,2],[1,4]]],[3,[2,3,4],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[2,1],[[-3,1],[2,1],[7,1]],[[1,4],[1,2],[1,4]]],[[3,1],[[2,1],[3,1],[4,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0041","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[3,[2,3,4],[[1,4],[1,2],[1,4]]],[4,[3,4,5],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145041,"parameters":{"laws":[[3,[2,3,4],[[1,4],[1,2],[1,4]]],[4,[3,4,5],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[3,1],[[2,1],[3,1],[4,1]],[[1,4],[1,2],[1,4]]],[[4,1],[[3,1],[4,1],[5,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0042","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[4,[2,4,6],[[1,4],[1,2],[1,4]]],[5,[4,5,6],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[4, [[2, 1/4], [4, 1/2], [6, 1/4]], 2], [5, [[4, 1/4], [5, 1/2], [6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[4, [[2, 1/4], [4, 1/2], [6, 1/4]], 2], [5, [[4, 1/4], [5, 1/2], [6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145042,"parameters":{"laws":[[4,[2,4,6],[[1,4],[1,2],[1,4]]],[5,[4,5,6],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[4,1],[[2,1],[4,1],[6,1]],[[1,4],[1,2],[1,4]]],[[5,1],[[4,1],[5,1],[6,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0043","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[5,[2,5,8],[[1,4],[1,2],[1,4]]],[6,[5,6,7],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[5, [[2, 1/4], [5, 1/2], [8, 1/4]], 9/2], [6, [[5, 1/4], [6, 1/2], [7, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[5, [[2, 1/4], [5, 1/2], [8, 1/4]], 9/2], [6, [[5, 1/4], [6, 1/2], [7, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145043,"parameters":{"laws":[[5,[2,5,8],[[1,4],[1,2],[1,4]]],[6,[5,6,7],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[5,1],[[2,1],[5,1],[8,1]],[[1,4],[1,2],[1,4]]],[[6,1],[[5,1],[6,1],[7,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0044","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[6,[2,6,10],[[1,4],[1,2],[1,4]]],[7,[6,7,8],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[6, [[2, 1/4], [6, 1/2], [10, 1/4]], 8], [7, [[6, 1/4], [7, 1/2], [8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[6, [[2, 1/4], [6, 1/2], [10, 1/4]], 8], [7, [[6, 1/4], [7, 1/2], [8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145044,"parameters":{"laws":[[6,[2,6,10],[[1,4],[1,2],[1,4]]],[7,[6,7,8],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[6,1],[[10,1],[2,1],[6,1]],[[1,4],[1,4],[1,2]]],[[7,1],[[6,1],[7,1],[8,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0045","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[7,[2,7,12],[[1,4],[1,2],[1,4]]],[8,[7,8,9],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[7, [[2, 1/4], [7, 1/2], [12, 1/4]], 25/2], [8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[7, [[2, 1/4], [7, 1/2], [12, 1/4]], 25/2], [8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145045,"parameters":{"laws":[[7,[2,7,12],[[1,4],[1,2],[1,4]]],[8,[7,8,9],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[7,1],[[12,1],[2,1],[7,1]],[[1,4],[1,4],[1,2]]],[[8,1],[[7,1],[8,1],[9,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0046","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[8,[7,8,9],[[1,4],[1,2],[1,4]]],[9,[8,9,10],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2], [9, [[8, 1/4], [9, 1/2], [10, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2], [9, [[8, 1/4], [9, 1/2], [10, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145046,"parameters":{"laws":[[8,[7,8,9],[[1,4],[1,2],[1,4]]],[9,[8,9,10],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[8,1],[[7,1],[8,1],[9,1]],[[1,4],[1,2],[1,4]]],[[9,1],[[10,1],[8,1],[9,1]],[[1,4],[1,4],[1,2]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0047","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[9,[7,9,11],[[1,4],[1,2],[1,4]]],[10,[9,10,11],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[10, [[9, 1/4], [10, 1/2], [11, 1/4]], 1/2], [9, [[7, 1/4], [9, 1/2], [11, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[10, [[9, 1/4], [10, 1/2], [11, 1/4]], 1/2], [9, [[7, 1/4], [9, 1/2], [11, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145047,"parameters":{"laws":[[9,[7,9,11],[[1,4],[1,2],[1,4]]],[10,[9,10,11],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[10,1],[[10,1],[11,1],[9,1]],[[1,2],[1,4],[1,4]]],[[9,1],[[11,1],[7,1],[9,1]],[[1,4],[1,4],[1,2]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0048","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[10,[7,10,13],[[1,4],[1,2],[1,4]]],[11,[10,11,12],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[10, [[7, 1/4], [10, 1/2], [13, 1/4]], 9/2], [11, [[10, 1/4], [11, 1/2], [12, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[10, [[7, 1/4], [10, 1/2], [13, 1/4]], 9/2], [11, [[10, 1/4], [11, 1/2], [12, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145048,"parameters":{"laws":[[10,[7,10,13],[[1,4],[1,2],[1,4]]],[11,[10,11,12],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[10,1],[[10,1],[13,1],[7,1]],[[1,2],[1,4],[1,4]]],[[11,1],[[10,1],[11,1],[12,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0049","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[11,[7,11,15],[[1,4],[1,2],[1,4]]],[12,[11,12,13],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[11, [[7, 1/4], [11, 1/2], [15, 1/4]], 8], [12, [[11, 1/4], [12, 1/2], [13, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[11, [[7, 1/4], [11, 1/2], [15, 1/4]], 8], [12, [[11, 1/4], [12, 1/2], [13, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145049,"parameters":{"laws":[[11,[7,11,15],[[1,4],[1,2],[1,4]]],[12,[11,12,13],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[11,1],[[11,1],[15,1],[7,1]],[[1,2],[1,4],[1,4]]],[[12,1],[[11,1],[12,1],[13,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0050","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-multi-theta-table；資料={\"laws\":[[12,[7,12,17],[[1,4],[1,2],[1,4]]],[13,[12,13,14],[[1,4],[1,2],[1,4]]]]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"([[12, [[7, 1/4], [12, 1/2], [17, 1/4]], 25/2], [13, [[12, 1/4], [13, 1/2], [14, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation_zh":"Wave6第八層核對=([[12, [[7, 1/4], [12, 1/2], [17, 1/4]], 25/2], [13, [[12, 1/4], [13, 1/2], [14, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-multi-theta-table","seed":202611145050,"parameters":{"laws":[[12,[7,12,17],[[1,4],[1,2],[1,4]]],[13,[12,13,14],[[1,4],[1,2],[1,4]]]],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-multi-theta-table","semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[12,1],[[12,1],[17,1],[7,1]],[[1,2],[1,4],[1,4]]],[[13,1],[[12,1],[13,1],[14,1]],[[1,4],[1,2],[1,4]]]]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0051","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-12,\"support\":[-12,-11],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145051,"parameters":{"theta":-12,"support":[-12,-11],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-11,1],[-12,1]],\"theta\":[-12,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0052","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-11,\"support\":[-11,-9],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, [[-11, 1, 0, 0], [-9, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-11, [[-11, 1, 0, 0], [-9, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145052,"parameters":{"theta":-11,"support":[-11,-9],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-11,1],[-9,1]],\"theta\":[-11,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0053","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-10,\"support\":[-10,-7],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, [[-10, 1, 0, 0], [-7, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-10, [[-10, 1, 0, 0], [-7, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145053,"parameters":{"theta":-10,"support":[-10,-7],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-10,1],[-7,1]],\"theta\":[-10,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0054","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-9,\"support\":[-9,-5],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, [[-9, 1, 0, 0], [-5, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-9, [[-9, 1, 0, 0], [-5, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145054,"parameters":{"theta":-9,"support":[-9,-5],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-5,1],[-9,1]],\"theta\":[-9,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0055","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-8,\"support\":[-8,-3],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, [[-8, 1, 0, 0], [-3, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-8, [[-8, 1, 0, 0], [-3, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145055,"parameters":{"theta":-8,"support":[-8,-3],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-3,1],[-8,1]],\"theta\":[-8,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0056","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-7,\"support\":[-7,-6],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145056,"parameters":{"theta":-7,"support":[-7,-6],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-6,1],[-7,1]],\"theta\":[-7,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0057","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-6,\"support\":[-6,-4],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, [[-6, 1, 0, 0], [-4, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-6, [[-6, 1, 0, 0], [-4, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145057,"parameters":{"theta":-6,"support":[-6,-4],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-4,1],[-6,1]],\"theta\":[-6,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0058","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-5,\"support\":[-5,-2],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, [[-5, 1, 0, 0], [-2, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-5, [[-5, 1, 0, 0], [-2, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145058,"parameters":{"theta":-5,"support":[-5,-2],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-2,1],[-5,1]],\"theta\":[-5,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0059","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-4,\"support\":[-4,0],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, [[-4, 1, 0, 0], [0, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-4, [[-4, 1, 0, 0], [0, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145059,"parameters":{"theta":-4,"support":[-4,0],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-4,1],[0,1]],\"theta\":[-4,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0060","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-3,\"support\":[-3,2],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, [[-3, 1, 0, 0], [2, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-3, [[-3, 1, 0, 0], [2, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145060,"parameters":{"theta":-3,"support":[-3,2],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-3,1],[2,1]],\"theta\":[-3,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0061","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-2,\"support\":[-2,-1],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, [[-2, 1, 0, 0], [-1, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-2, [[-2, 1, 0, 0], [-1, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145061,"parameters":{"theta":-2,"support":[-2,-1],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-1,1],[-2,1]],\"theta\":[-2,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0062","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":-1,\"support\":[-1,1],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, [[-1, 1, 0, 0], [1, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(-1, [[-1, 1, 0, 0], [1, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145062,"parameters":{"theta":-1,"support":[-1,1],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-1,1],[1,1]],\"theta\":[-1,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0063","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":0,\"support\":[0,3],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, [[0, 1, 0, 0], [3, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(0, [[0, 1, 0, 0], [3, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145063,"parameters":{"theta":0,"support":[0,3],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[0,1],[3,1]],\"theta\":[0,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0064","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":1,\"support\":[1,5],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, [[1, 1, 0, 0], [5, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(1, [[1, 1, 0, 0], [5, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145064,"parameters":{"theta":1,"support":[1,5],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[1,1],[5,1]],\"theta\":[1,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0065","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":2,\"support\":[2,7],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, [[2, 1, 0, 0], [7, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(2, [[2, 1, 0, 0], [7, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145065,"parameters":{"theta":2,"support":[2,7],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[2,1],[7,1]],\"theta\":[2,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0066","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":3,\"support\":[3,4],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, [[3, 1, 0, 0], [4, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(3, [[3, 1, 0, 0], [4, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145066,"parameters":{"theta":3,"support":[3,4],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[3,1],[4,1]],\"theta\":[3,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0067","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":4,\"support\":[4,6],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, [[4, 1, 0, 0], [6, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(4, [[4, 1, 0, 0], [6, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145067,"parameters":{"theta":4,"support":[4,6],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[4,1],[6,1]],\"theta\":[4,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0068","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":5,\"support\":[5,8],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, [[5, 1, 0, 0], [8, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(5, [[5, 1, 0, 0], [8, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145068,"parameters":{"theta":5,"support":[5,8],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[5,1],[8,1]],\"theta\":[5,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0069","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":6,\"support\":[6,10],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, [[6, 1, 0, 0], [10, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(6, [[6, 1, 0, 0], [10, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145069,"parameters":{"theta":6,"support":[6,10],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[10,1],[6,1]],\"theta\":[6,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0070","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":7,\"support\":[7,12],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, [[7, 1, 0, 0], [12, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(7, [[7, 1, 0, 0], [12, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145070,"parameters":{"theta":7,"support":[7,12],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[12,1],[7,1]],\"theta\":[7,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0071","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":8,\"support\":[8,9],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, [[8, 1, 0, 0], [9, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(8, [[8, 1, 0, 0], [9, 0, 1, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145071,"parameters":{"theta":8,"support":[8,9],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[8,1],[9,1]],\"theta\":[8,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0072","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":9,\"support\":[9,11],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, [[9, 1, 0, 0], [11, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(9, [[9, 1, 0, 0], [11, 0, 4, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145072,"parameters":{"theta":9,"support":[9,11],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[11,1],[9,1]],\"theta\":[9,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0073","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":10,\"support\":[10,13],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, [[10, 1, 0, 0], [13, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(10, [[10, 1, 0, 0], [13, 0, 9, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145073,"parameters":{"theta":10,"support":[10,13],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[10,1],[13,1]],\"theta\":[10,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0074","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":11,\"support\":[11,15],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, [[11, 1, 0, 0], [15, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(11, [[11, 1, 0, 0], [15, 0, 16, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145074,"parameters":{"theta":11,"support":[11,15],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[11,1],[15,1]],\"theta\":[11,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0075","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-zero-iff-almost-sure；資料={\"theta\":12,\"support\":[12,17],\"probabilities\":[1,0]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, [[12, 1, 0, 0], [17, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation_zh":"Wave6第八層核對=(12, [[12, 1, 0, 0], [17, 0, 25, 0]], 0, 是, 是, 是, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145075,"parameters":{"theta":12,"support":[12,17],"probabilities":[1,0],"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-zero-iff-almost-sure","semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[12,1],[17,1]],\"theta\":[12,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0076","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-12,\"support\":[-13,-12,-11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, -13, 1, -12, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-12, -13, 1, -12, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145076,"parameters":{"theta":-12,"support":[-13,-12,-11],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-11,1],[-12,1],[-13,1]],\"theta\":[-12,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0077","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-11,\"support\":[-13,-11,-9],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, -13, 4, -11, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-11, -13, 4, -11, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145077,"parameters":{"theta":-11,"support":[-13,-11,-9],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-11,1],[-13,1],[-9,1]],\"theta\":[-11,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0078","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-10,\"support\":[-13,-10,-7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, -13, 9, -10, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-10, -13, 9, -10, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145078,"parameters":{"theta":-10,"support":[-13,-10,-7],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0079","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-9,\"support\":[-13,-9,-5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, -13, 16, -9, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-9, -13, 16, -9, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145079,"parameters":{"theta":-9,"support":[-13,-9,-5],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-5,1],[-9,1]],\"theta\":[-9,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0080","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-8,\"support\":[-13,-8,-3],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, -13, 25, -8, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-8, -13, 25, -8, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145080,"parameters":{"theta":-8,"support":[-13,-8,-3],"probabilities":[[1,4],[1,2],[1,4]],"observed":-13,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-3,1],[-8,1]],\"theta\":[-8,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0081","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-7,\"support\":[-8,-7,-6],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, -8, 1, -7, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-7, -8, 1, -7, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145081,"parameters":{"theta":-7,"support":[-8,-7,-6],"probabilities":[[1,4],[1,2],[1,4]],"observed":-8,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-6,1],[-7,1],[-8,1]],\"theta\":[-7,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0082","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-6,\"support\":[-8,-6,-4],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, -8, 4, -6, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-6, -8, 4, -6, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145082,"parameters":{"theta":-6,"support":[-8,-6,-4],"probabilities":[[1,4],[1,2],[1,4]],"observed":-8,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-6,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0083","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-5,\"support\":[-8,-5,-2],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, -8, 9, -5, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-5, -8, 9, -5, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145083,"parameters":{"theta":-5,"support":[-8,-5,-2],"probabilities":[[1,4],[1,2],[1,4]],"observed":-8,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-2,1],[-5,1],[-8,1]],\"theta\":[-5,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0084","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-4,\"support\":[-8,-4,0],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, -8, 16, -4, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-4, -8, 16, -4, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145084,"parameters":{"theta":-4,"support":[-8,-4,0],"probabilities":[[1,4],[1,2],[1,4]],"observed":-8,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-4,1],[-8,1],[0,1]],\"theta\":[-4,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0085","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-3,\"support\":[-8,-3,2],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, -8, 25, -3, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-3, -8, 25, -3, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145085,"parameters":{"theta":-3,"support":[-8,-3,2],"probabilities":[[1,4],[1,2],[1,4]],"observed":-8,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-3,1],[-8,1],[2,1]],\"theta\":[-3,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0086","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-2,\"support\":[-3,-2,-1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, -3, 1, -2, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-2, -3, 1, -2, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145086,"parameters":{"theta":-2,"support":[-3,-2,-1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-3,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-2,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0087","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":-1,\"support\":[-3,-1,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, -3, 4, -1, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(-1, -3, 4, -1, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145087,"parameters":{"theta":-1,"support":[-3,-1,1],"probabilities":[[1,4],[1,2],[1,4]],"observed":-3,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-1,1],[-3,1],[1,1]],\"theta\":[-1,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0088","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":0,\"support\":[-3,0,3],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, -3, 9, 0, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(0, -3, 9, 0, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145088,"parameters":{"theta":0,"support":[-3,0,3],"probabilities":[[1,4],[1,2],[1,4]],"observed":-3,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[0,1],[3,1]],\"theta\":[0,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0089","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":1,\"support\":[-3,1,5],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, -3, 16, 1, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(1, -3, 16, 1, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145089,"parameters":{"theta":1,"support":[-3,1,5],"probabilities":[[1,4],[1,2],[1,4]],"observed":-3,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[1,1],[5,1]],\"theta\":[1,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0090","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":2,\"support\":[-3,2,7],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, -3, 25, 2, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(2, -3, 25, 2, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145090,"parameters":{"theta":2,"support":[-3,2,7],"probabilities":[[1,4],[1,2],[1,4]],"observed":-3,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[2,1],[7,1]],\"theta\":[2,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0091","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":3,\"support\":[2,3,4],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 2, 1, 3, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(3, 2, 1, 3, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145091,"parameters":{"theta":3,"support":[2,3,4],"probabilities":[[1,4],[1,2],[1,4]],"observed":2,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[3,1],[4,1]],\"theta\":[3,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0092","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":4,\"support\":[2,4,6],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 2, 4, 4, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(4, 2, 4, 4, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145092,"parameters":{"theta":4,"support":[2,4,6],"probabilities":[[1,4],[1,2],[1,4]],"observed":2,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[4,1],[6,1]],\"theta\":[4,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0093","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":5,\"support\":[2,5,8],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 2, 9, 5, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(5, 2, 9, 5, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145093,"parameters":{"theta":5,"support":[2,5,8],"probabilities":[[1,4],[1,2],[1,4]],"observed":2,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[5,1],[8,1]],\"theta\":[5,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0094","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":6,\"support\":[2,6,10],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 2, 16, 6, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(6, 2, 16, 6, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145094,"parameters":{"theta":6,"support":[2,6,10],"probabilities":[[1,4],[1,2],[1,4]],"observed":2,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[10,1],[2,1],[6,1]],\"theta\":[6,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0095","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":7,\"support\":[2,7,12],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 2, 25, 7, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(7, 2, 25, 7, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145095,"parameters":{"theta":7,"support":[2,7,12],"probabilities":[[1,4],[1,2],[1,4]],"observed":2,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[12,1],[2,1],[7,1]],\"theta\":[7,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0096","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":8,\"support\":[7,8,9],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 7, 1, 8, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(8, 7, 1, 8, 是, 1/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145096,"parameters":{"theta":8,"support":[7,8,9],"probabilities":[[1,4],[1,2],[1,4]],"observed":7,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[7,1],[8,1],[9,1]],\"theta\":[8,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0097","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":9,\"support\":[7,9,11],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 7, 4, 9, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(9, 7, 4, 9, 是, 2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145097,"parameters":{"theta":9,"support":[7,9,11],"probabilities":[[1,4],[1,2],[1,4]],"observed":7,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[7,1],[9,1]],\"theta\":[9,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0098","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":10,\"support\":[7,10,13],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 7, 9, 10, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(10, 7, 9, 10, 是, 9/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145098,"parameters":{"theta":10,"support":[7,10,13],"probabilities":[[1,4],[1,2],[1,4]],"observed":7,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[13,1],[7,1]],\"theta\":[10,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0099","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":11,\"support\":[7,11,15],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 7, 16, 11, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(11, 7, 16, 11, 是, 8, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145099,"parameters":{"theta":11,"support":[7,11,15],"probabilities":[[1,4],[1,2],[1,4]],"observed":7,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[11,1],[15,1],[7,1]],\"theta\":[11,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0100","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","language":"zh-Hant","source_mko_ids":["mko-estimator-mean-squared-error"],"title_zh":"估計量均方誤差","stem_zh":"scalar theta下完整finite sampling law的exact expected squared error。操作=mse-observed-unbiased-boundary；資料={\"theta\":12,\"support\":[7,12,17],\"probabilities\":[[1,4],[1,2],[1,4]],\"observed\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 7, 25, 12, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation_zh":"Wave6第八層核對=(12, 7, 25, 12, 是, 25/2, 否, 是, 否, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)；依完整明示資料獨立重算；MSE逐theta直接加權平方誤差；MSE=0恰當T=theta almost surely；單一observed error不是MSE；unbiased不推出zero MSE，亦不使用bias-variance decomposition或推出consistency。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["估計量均方誤差"]},"variant":{"family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145100,"parameters":{"theta":12,"support":[7,12,17],"probabilities":[[1,4],[1,2],[1,4]],"observed":7,"domain":"Exact_finite_scalar_estimator_MSE","operation":"mse-observed-unbiased-boundary","semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[12,1],[17,1],[7,1]],\"theta\":[12,1]}"}},"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、非同態kernel、未驗normal的quotient、class-size公式、finite-quadrature Gaussian定理、bias-variance decomposition、LP最優性、minimum vertex cover、Lean或Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-15","contract_version":"ocme-question-factory-v0.1"}}