{"schema_version":"ocme-question-v0.1","id":"q-20261015-advance-20261114-r04-standard-normal-distribution-0001","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, (2*pi)^(-1/2)*exp(-1/8), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(1/2, (2*pi)^(-1/2)*exp(-1/8), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144001,"parameters":{"x":[1,2],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[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-r04-standard-normal-distribution-0002","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2/3, (2*pi)^(-1/2)*exp(-2/9), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-2/3, (2*pi)^(-1/2)*exp(-2/9), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144002,"parameters":{"x":[-2,3],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0003","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3/4, (2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(3/4, (2*pi)^(-1/2)*exp(-9/32), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144003,"parameters":{"x":[3,4],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0004","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4/5, (2*pi)^(-1/2)*exp(-8/25), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-4/5, (2*pi)^(-1/2)*exp(-8/25), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144004,"parameters":{"x":[-4,5],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0005","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5/6, (2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(5/6, (2*pi)^(-1/2)*exp(-25/72), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144005,"parameters":{"x":[5,6],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0006","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6/7, (2*pi)^(-1/2)*exp(-18/49), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-6/7, (2*pi)^(-1/2)*exp(-18/49), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144006,"parameters":{"x":[-6,7],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0007","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7/8, (2*pi)^(-1/2)*exp(-49/128), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(7/8, (2*pi)^(-1/2)*exp(-49/128), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144007,"parameters":{"x":[7,8],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0008","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8/9, (2*pi)^(-1/2)*exp(-32/81), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-8/9, (2*pi)^(-1/2)*exp(-32/81), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144008,"parameters":{"x":[-8,9],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0009","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9/10, (2*pi)^(-1/2)*exp(-81/200), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(9/10, (2*pi)^(-1/2)*exp(-81/200), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144009,"parameters":{"x":[9,10],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0010","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10/11, (2*pi)^(-1/2)*exp(-50/121), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-10/11, (2*pi)^(-1/2)*exp(-50/121), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144010,"parameters":{"x":[-10,11],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0011","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11/12, (2*pi)^(-1/2)*exp(-121/288), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(11/12, (2*pi)^(-1/2)*exp(-121/288), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144011,"parameters":{"x":[11,12],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0012","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12/13, (2*pi)^(-1/2)*exp(-72/169), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-12/13, (2*pi)^(-1/2)*exp(-72/169), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144012,"parameters":{"x":[-12,13],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0013","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13/14, (2*pi)^(-1/2)*exp(-169/392), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(13/14, (2*pi)^(-1/2)*exp(-169/392), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144013,"parameters":{"x":[13,14],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0014","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14/15, (2*pi)^(-1/2)*exp(-98/225), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-14/15, (2*pi)^(-1/2)*exp(-98/225), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144014,"parameters":{"x":[-14,15],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0015","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15/16, (2*pi)^(-1/2)*exp(-225/512), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(15/16, (2*pi)^(-1/2)*exp(-225/512), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144015,"parameters":{"x":[15,16],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0016","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16/17, (2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-16/17, (2*pi)^(-1/2)*exp(-128/289), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144016,"parameters":{"x":[-16,17],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0017","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17/18, (2*pi)^(-1/2)*exp(-289/648), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(17/18, (2*pi)^(-1/2)*exp(-289/648), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144017,"parameters":{"x":[17,18],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0018","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18/19, (2*pi)^(-1/2)*exp(-162/361), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-18/19, (2*pi)^(-1/2)*exp(-162/361), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144018,"parameters":{"x":[-18,19],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0019","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19/20, (2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(19/20, (2*pi)^(-1/2)*exp(-361/800), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144019,"parameters":{"x":[19,20],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0020","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20/21, (2*pi)^(-1/2)*exp(-200/441), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-20/21, (2*pi)^(-1/2)*exp(-200/441), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144020,"parameters":{"x":[-20,21],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0021","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21/22, (2*pi)^(-1/2)*exp(-441/968), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(21/22, (2*pi)^(-1/2)*exp(-441/968), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144021,"parameters":{"x":[21,22],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0022","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22/23, (2*pi)^(-1/2)*exp(-242/529), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-22/23, (2*pi)^(-1/2)*exp(-242/529), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144022,"parameters":{"x":[-22,23],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0023","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23/24, (2*pi)^(-1/2)*exp(-529/1152), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(23/24, (2*pi)^(-1/2)*exp(-529/1152), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144023,"parameters":{"x":[23,24],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0024","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[-24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24/25, (2*pi)^(-1/2)*exp(-288/625), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(-24/25, (2*pi)^(-1/2)*exp(-288/625), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144024,"parameters":{"x":[-24,25],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[-24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0025","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-symbolic-density；資料={\"x\":[25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25/26, (2*pi)^(-1/2)*exp(-625/1352), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)","explanation_zh":"Wave6第八層核對=(25/26, (2*pi)^(-1/2)*exp(-625/1352), phi(x)=(2*pi)^(-1/2)*exp(-x^2/2), normalization_is_a_named_Gaussian_integral_fact_not_a_finite_evaluation)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-symbolic-density","seed":202611144025,"parameters":{"x":[25,26],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-symbolic-density","semantic_template":"qf-mod45-normaldist-symbolic-density/{\"x\":[25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0026","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[1,2]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1/2, -1/2, (2*pi)^(-1/2)*exp(-1/8), (2*pi)^(-1/2)*exp(-1/8), 是, phi(x)/phi(0)=exp(-1/8), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(1/2, -1/2, (2*pi)^(-1/2)*exp(-1/8), (2*pi)^(-1/2)*exp(-1/8), 是, phi(x)/phi(0)=exp(-1/8), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144026,"parameters":{"x":[1,2],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[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-r04-standard-normal-distribution-0027","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-2,3]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2/3, 2/3, (2*pi)^(-1/2)*exp(-2/9), (2*pi)^(-1/2)*exp(-2/9), 是, phi(x)/phi(0)=exp(-2/9), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-2/3, 2/3, (2*pi)^(-1/2)*exp(-2/9), (2*pi)^(-1/2)*exp(-2/9), 是, phi(x)/phi(0)=exp(-2/9), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144027,"parameters":{"x":[-2,3],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-2,3]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0028","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[3,4]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), 是, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(3/4, -3/4, (2*pi)^(-1/2)*exp(-9/32), (2*pi)^(-1/2)*exp(-9/32), 是, phi(x)/phi(0)=exp(-9/32), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144028,"parameters":{"x":[3,4],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[3,4]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0029","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-4,5]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4/5, 4/5, (2*pi)^(-1/2)*exp(-8/25), (2*pi)^(-1/2)*exp(-8/25), 是, phi(x)/phi(0)=exp(-8/25), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-4/5, 4/5, (2*pi)^(-1/2)*exp(-8/25), (2*pi)^(-1/2)*exp(-8/25), 是, phi(x)/phi(0)=exp(-8/25), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144029,"parameters":{"x":[-4,5],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-4,5]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0030","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[5,6]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(5/6, -5/6, (2*pi)^(-1/2)*exp(-25/72), (2*pi)^(-1/2)*exp(-25/72), 是, phi(x)/phi(0)=exp(-25/72), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144030,"parameters":{"x":[5,6],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[5,6]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0031","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-6,7]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6/7, 6/7, (2*pi)^(-1/2)*exp(-18/49), (2*pi)^(-1/2)*exp(-18/49), 是, phi(x)/phi(0)=exp(-18/49), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-6/7, 6/7, (2*pi)^(-1/2)*exp(-18/49), (2*pi)^(-1/2)*exp(-18/49), 是, phi(x)/phi(0)=exp(-18/49), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144031,"parameters":{"x":[-6,7],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-6,7]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0032","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[7,8]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7/8, -7/8, (2*pi)^(-1/2)*exp(-49/128), (2*pi)^(-1/2)*exp(-49/128), 是, phi(x)/phi(0)=exp(-49/128), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(7/8, -7/8, (2*pi)^(-1/2)*exp(-49/128), (2*pi)^(-1/2)*exp(-49/128), 是, phi(x)/phi(0)=exp(-49/128), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144032,"parameters":{"x":[7,8],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[7,8]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0033","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-8,9]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8/9, 8/9, (2*pi)^(-1/2)*exp(-32/81), (2*pi)^(-1/2)*exp(-32/81), 是, phi(x)/phi(0)=exp(-32/81), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-8/9, 8/9, (2*pi)^(-1/2)*exp(-32/81), (2*pi)^(-1/2)*exp(-32/81), 是, phi(x)/phi(0)=exp(-32/81), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144033,"parameters":{"x":[-8,9],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-8,9]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0034","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[9,10]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9/10, -9/10, (2*pi)^(-1/2)*exp(-81/200), (2*pi)^(-1/2)*exp(-81/200), 是, phi(x)/phi(0)=exp(-81/200), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(9/10, -9/10, (2*pi)^(-1/2)*exp(-81/200), (2*pi)^(-1/2)*exp(-81/200), 是, phi(x)/phi(0)=exp(-81/200), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144034,"parameters":{"x":[9,10],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[9,10]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0035","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-10,11]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10/11, 10/11, (2*pi)^(-1/2)*exp(-50/121), (2*pi)^(-1/2)*exp(-50/121), 是, phi(x)/phi(0)=exp(-50/121), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-10/11, 10/11, (2*pi)^(-1/2)*exp(-50/121), (2*pi)^(-1/2)*exp(-50/121), 是, phi(x)/phi(0)=exp(-50/121), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144035,"parameters":{"x":[-10,11],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-10,11]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0036","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[11,12]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11/12, -11/12, (2*pi)^(-1/2)*exp(-121/288), (2*pi)^(-1/2)*exp(-121/288), 是, phi(x)/phi(0)=exp(-121/288), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(11/12, -11/12, (2*pi)^(-1/2)*exp(-121/288), (2*pi)^(-1/2)*exp(-121/288), 是, phi(x)/phi(0)=exp(-121/288), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144036,"parameters":{"x":[11,12],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[11,12]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0037","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-12,13]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12/13, 12/13, (2*pi)^(-1/2)*exp(-72/169), (2*pi)^(-1/2)*exp(-72/169), 是, phi(x)/phi(0)=exp(-72/169), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-12/13, 12/13, (2*pi)^(-1/2)*exp(-72/169), (2*pi)^(-1/2)*exp(-72/169), 是, phi(x)/phi(0)=exp(-72/169), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144037,"parameters":{"x":[-12,13],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-12,13]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0038","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[13,14]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13/14, -13/14, (2*pi)^(-1/2)*exp(-169/392), (2*pi)^(-1/2)*exp(-169/392), 是, phi(x)/phi(0)=exp(-169/392), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(13/14, -13/14, (2*pi)^(-1/2)*exp(-169/392), (2*pi)^(-1/2)*exp(-169/392), 是, phi(x)/phi(0)=exp(-169/392), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144038,"parameters":{"x":[13,14],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[13,14]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0039","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-14,15]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-14/15, 14/15, (2*pi)^(-1/2)*exp(-98/225), (2*pi)^(-1/2)*exp(-98/225), 是, phi(x)/phi(0)=exp(-98/225), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-14/15, 14/15, (2*pi)^(-1/2)*exp(-98/225), (2*pi)^(-1/2)*exp(-98/225), 是, phi(x)/phi(0)=exp(-98/225), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144039,"parameters":{"x":[-14,15],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-14,15]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0040","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[15,16]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15/16, -15/16, (2*pi)^(-1/2)*exp(-225/512), (2*pi)^(-1/2)*exp(-225/512), 是, phi(x)/phi(0)=exp(-225/512), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(15/16, -15/16, (2*pi)^(-1/2)*exp(-225/512), (2*pi)^(-1/2)*exp(-225/512), 是, phi(x)/phi(0)=exp(-225/512), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144040,"parameters":{"x":[15,16],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[15,16]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0041","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-16,17]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-16/17, 16/17, (2*pi)^(-1/2)*exp(-128/289), (2*pi)^(-1/2)*exp(-128/289), 是, phi(x)/phi(0)=exp(-128/289), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-16/17, 16/17, (2*pi)^(-1/2)*exp(-128/289), (2*pi)^(-1/2)*exp(-128/289), 是, phi(x)/phi(0)=exp(-128/289), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144041,"parameters":{"x":[-16,17],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-16,17]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0042","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[17,18]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17/18, -17/18, (2*pi)^(-1/2)*exp(-289/648), (2*pi)^(-1/2)*exp(-289/648), 是, phi(x)/phi(0)=exp(-289/648), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(17/18, -17/18, (2*pi)^(-1/2)*exp(-289/648), (2*pi)^(-1/2)*exp(-289/648), 是, phi(x)/phi(0)=exp(-289/648), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144042,"parameters":{"x":[17,18],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[17,18]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0043","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-18,19]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-18/19, 18/19, (2*pi)^(-1/2)*exp(-162/361), (2*pi)^(-1/2)*exp(-162/361), 是, phi(x)/phi(0)=exp(-162/361), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-18/19, 18/19, (2*pi)^(-1/2)*exp(-162/361), (2*pi)^(-1/2)*exp(-162/361), 是, phi(x)/phi(0)=exp(-162/361), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144043,"parameters":{"x":[-18,19],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-18,19]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0044","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[19,20]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), 是, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(19/20, -19/20, (2*pi)^(-1/2)*exp(-361/800), (2*pi)^(-1/2)*exp(-361/800), 是, phi(x)/phi(0)=exp(-361/800), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144044,"parameters":{"x":[19,20],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[19,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0045","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-20,21]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-20/21, 20/21, (2*pi)^(-1/2)*exp(-200/441), (2*pi)^(-1/2)*exp(-200/441), 是, phi(x)/phi(0)=exp(-200/441), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-20/21, 20/21, (2*pi)^(-1/2)*exp(-200/441), (2*pi)^(-1/2)*exp(-200/441), 是, phi(x)/phi(0)=exp(-200/441), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144045,"parameters":{"x":[-20,21],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-20,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0046","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[21,22]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21/22, -21/22, (2*pi)^(-1/2)*exp(-441/968), (2*pi)^(-1/2)*exp(-441/968), 是, phi(x)/phi(0)=exp(-441/968), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(21/22, -21/22, (2*pi)^(-1/2)*exp(-441/968), (2*pi)^(-1/2)*exp(-441/968), 是, phi(x)/phi(0)=exp(-441/968), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144046,"parameters":{"x":[21,22],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[21,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0047","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-22,23]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-22/23, 22/23, (2*pi)^(-1/2)*exp(-242/529), (2*pi)^(-1/2)*exp(-242/529), 是, phi(x)/phi(0)=exp(-242/529), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-22/23, 22/23, (2*pi)^(-1/2)*exp(-242/529), (2*pi)^(-1/2)*exp(-242/529), 是, phi(x)/phi(0)=exp(-242/529), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144047,"parameters":{"x":[-22,23],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-22,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0048","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[23,24]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23/24, -23/24, (2*pi)^(-1/2)*exp(-529/1152), (2*pi)^(-1/2)*exp(-529/1152), 是, phi(x)/phi(0)=exp(-529/1152), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(23/24, -23/24, (2*pi)^(-1/2)*exp(-529/1152), (2*pi)^(-1/2)*exp(-529/1152), 是, phi(x)/phi(0)=exp(-529/1152), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144048,"parameters":{"x":[23,24],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[23,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0049","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[-24,25]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-24/25, 24/25, (2*pi)^(-1/2)*exp(-288/625), (2*pi)^(-1/2)*exp(-288/625), 是, phi(x)/phi(0)=exp(-288/625), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(-24/25, 24/25, (2*pi)^(-1/2)*exp(-288/625), (2*pi)^(-1/2)*exp(-288/625), 是, phi(x)/phi(0)=exp(-288/625), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144049,"parameters":{"x":[-24,25],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[-24,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0050","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-evenness-ratio；資料={\"x\":[25,26]}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25/26, -25/26, (2*pi)^(-1/2)*exp(-625/1352), (2*pi)^(-1/2)*exp(-625/1352), 是, phi(x)/phi(0)=exp(-625/1352), evenness_follows_from_x_squared)","explanation_zh":"Wave6第八層核對=(25/26, -25/26, (2*pi)^(-1/2)*exp(-625/1352), (2*pi)^(-1/2)*exp(-625/1352), 是, phi(x)/phi(0)=exp(-625/1352), evenness_follows_from_x_squared)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-evenness-ratio","seed":202611144050,"parameters":{"x":[25,26],"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-evenness-ratio","semantic_template":"qf-mod45-normaldist-evenness-ratio/{\"x\":[25,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_or_symbolic_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"候選須主端重算；不構成人工審定、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-r04-standard-normal-distribution-0051","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-12, Phi(12)=1-Phi(-12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-12, Phi(12)=1-Phi(-12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144051,"parameters":{"x":-12,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0052","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-11, Phi(11)=1-Phi(-11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-11, Phi(11)=1-Phi(-11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144052,"parameters":{"x":-11,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0053","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-10, Phi(10)=1-Phi(-10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-10, Phi(10)=1-Phi(-10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144053,"parameters":{"x":-10,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0054","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-9, Phi(9)=1-Phi(-9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-9, Phi(9)=1-Phi(-9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144054,"parameters":{"x":-9,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0055","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-8, Phi(8)=1-Phi(-8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-8, Phi(8)=1-Phi(-8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144055,"parameters":{"x":-8,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0056","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-7, Phi(7)=1-Phi(-7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-7, Phi(7)=1-Phi(-7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144056,"parameters":{"x":-7,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0057","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-6, Phi(6)=1-Phi(-6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-6, Phi(6)=1-Phi(-6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144057,"parameters":{"x":-6,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0058","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-5, Phi(5)=1-Phi(-5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-5, Phi(5)=1-Phi(-5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144058,"parameters":{"x":-5,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0059","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-4, Phi(4)=1-Phi(-4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-4, Phi(4)=1-Phi(-4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144059,"parameters":{"x":-4,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0060","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-3, Phi(3)=1-Phi(-3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-3, Phi(3)=1-Phi(-3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144060,"parameters":{"x":-3,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0061","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-2, Phi(2)=1-Phi(-2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-2, Phi(2)=1-Phi(-2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144061,"parameters":{"x":-2,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0062","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":-1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(-1, Phi(1)=1-Phi(-1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(-1, Phi(1)=1-Phi(-1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144062,"parameters":{"x":-1,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[-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-r04-standard-normal-distribution-0063","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":0}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(0, Phi(0)=1-Phi(0), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(0, Phi(0)=1-Phi(0), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144063,"parameters":{"x":0,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0064","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":1}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, Phi(-1)=1-Phi(1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(1, Phi(-1)=1-Phi(1), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144064,"parameters":{"x":1,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0065","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":2}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, Phi(-2)=1-Phi(2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(2, Phi(-2)=1-Phi(2), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144065,"parameters":{"x":2,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0066","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":3}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, Phi(-3)=1-Phi(3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(3, Phi(-3)=1-Phi(3), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144066,"parameters":{"x":3,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0067","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":4}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, Phi(-4)=1-Phi(4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(4, Phi(-4)=1-Phi(4), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144067,"parameters":{"x":4,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0068","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":5}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, Phi(-5)=1-Phi(5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(5, Phi(-5)=1-Phi(5), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144068,"parameters":{"x":5,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0069","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":6}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, Phi(-6)=1-Phi(6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(6, Phi(-6)=1-Phi(6), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144069,"parameters":{"x":6,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0070","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":7}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, Phi(-7)=1-Phi(7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(7, Phi(-7)=1-Phi(7), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144070,"parameters":{"x":7,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0071","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":8}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, Phi(-8)=1-Phi(8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(8, Phi(-8)=1-Phi(8), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144071,"parameters":{"x":8,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0072","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":9}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, Phi(-9)=1-Phi(9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(9, Phi(-9)=1-Phi(9), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144072,"parameters":{"x":9,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0073","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, Phi(-10)=1-Phi(10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(10, Phi(-10)=1-Phi(10), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144073,"parameters":{"x":10,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0074","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, Phi(-11)=1-Phi(11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(11, Phi(-11)=1-Phi(11), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144074,"parameters":{"x":11,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0075","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-cdf-symmetry；資料={\"x\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, Phi(-12)=1-Phi(12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)","explanation_zh":"Wave6第八層核對=(12, Phi(-12)=1-Phi(12), Phi(0)=1/2, 是, declared_normalization_plus_even_density_implies_CDF_symmetry_but_no_nonzero_elementary_CDF_value)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-cdf-symmetry","seed":202611144075,"parameters":{"x":12,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-cdf-symmetry","semantic_template":"qf-mod45-normaldist-cdf-symmetry/{\"x\":[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-r04-standard-normal-distribution-0076","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":1,\"panels\":10}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(1, 10, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(1, 10, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144076,"parameters":{"window":1,"panels":10,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":10,\"window\":[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-r04-standard-normal-distribution-0077","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":2,\"panels\":11}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(2, 11, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(2, 11, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144077,"parameters":{"window":2,"panels":11,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":11,\"window\":[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-r04-standard-normal-distribution-0078","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":3,\"panels\":12}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(3, 12, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(3, 12, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144078,"parameters":{"window":3,"panels":12,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":12,\"window\":[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-r04-standard-normal-distribution-0079","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":4,\"panels\":13}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(4, 13, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(4, 13, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144079,"parameters":{"window":4,"panels":13,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":13,\"window\":[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-r04-standard-normal-distribution-0080","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":5,\"panels\":14}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(5, 14, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(5, 14, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144080,"parameters":{"window":5,"panels":14,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":14,\"window\":[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-r04-standard-normal-distribution-0081","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":6,\"panels\":15}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(6, 15, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(6, 15, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144081,"parameters":{"window":6,"panels":15,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":15,\"window\":[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-r04-standard-normal-distribution-0082","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":7,\"panels\":16}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(7, 16, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(7, 16, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144082,"parameters":{"window":7,"panels":16,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":16,\"window\":[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-r04-standard-normal-distribution-0083","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":8,\"panels\":17}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(8, 17, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(8, 17, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144083,"parameters":{"window":8,"panels":17,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":17,\"window\":[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-r04-standard-normal-distribution-0084","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":9,\"panels\":18}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(9, 18, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144084,"parameters":{"window":9,"panels":18,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":18,\"window\":[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-r04-standard-normal-distribution-0085","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":10,\"panels\":19}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(10, 19, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(10, 19, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144085,"parameters":{"window":10,"panels":19,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":19,\"window\":[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-r04-standard-normal-distribution-0086","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":11,\"panels\":20}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(11, 20, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(11, 20, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144086,"parameters":{"window":11,"panels":20,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":20,\"window\":[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-r04-standard-normal-distribution-0087","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":12,\"panels\":21}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(12, 21, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(12, 21, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144087,"parameters":{"window":12,"panels":21,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":21,\"window\":[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-r04-standard-normal-distribution-0088","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":13,\"panels\":22}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(13, 22, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(13, 22, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144088,"parameters":{"window":13,"panels":22,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":22,\"window\":[13,1]}"}},"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-r04-standard-normal-distribution-0089","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":14,\"panels\":23}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(14, 23, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(14, 23, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144089,"parameters":{"window":14,"panels":23,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":23,\"window\":[14,1]}"}},"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-r04-standard-normal-distribution-0090","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":15,\"panels\":24}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(15, 24, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(15, 24, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144090,"parameters":{"window":15,"panels":24,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":24,\"window\":[15,1]}"}},"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-r04-standard-normal-distribution-0091","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":16,\"panels\":25}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(16, 25, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(16, 25, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144091,"parameters":{"window":16,"panels":25,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":25,\"window\":[16,1]}"}},"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-r04-standard-normal-distribution-0092","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":17,\"panels\":26}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(17, 26, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(17, 26, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144092,"parameters":{"window":17,"panels":26,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":26,\"window\":[17,1]}"}},"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-r04-standard-normal-distribution-0093","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":18,\"panels\":27}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(18, 27, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(18, 27, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144093,"parameters":{"window":18,"panels":27,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":27,\"window\":[18,1]}"}},"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-r04-standard-normal-distribution-0094","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":19,\"panels\":28}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(19, 28, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(19, 28, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144094,"parameters":{"window":19,"panels":28,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":28,\"window\":[19,1]}"}},"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-r04-standard-normal-distribution-0095","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":20,\"panels\":29}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(20, 29, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(20, 29, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144095,"parameters":{"window":20,"panels":29,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":29,\"window\":[20,1]}"}},"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-r04-standard-normal-distribution-0096","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":21,\"panels\":30}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(21, 30, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(21, 30, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144096,"parameters":{"window":21,"panels":30,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":30,\"window\":[21,1]}"}},"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-r04-standard-normal-distribution-0097","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":22,\"panels\":31}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(22, 31, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(22, 31, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144097,"parameters":{"window":22,"panels":31,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":31,\"window\":[22,1]}"}},"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-r04-standard-normal-distribution-0098","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":23,\"panels\":32}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(23, 32, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(23, 32, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144098,"parameters":{"window":23,"panels":32,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":32,\"window\":[23,1]}"}},"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-r04-standard-normal-distribution-0099","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":24,\"panels\":33}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(24, 33, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(24, 33, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144099,"parameters":{"window":24,"panels":33,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":33,\"window\":[24,1]}"}},"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-r04-standard-normal-distribution-0100","batch_id":"batch-20261015-advance-20261114-r04-standard-normal-distribution-foundations","language":"zh-Hant","source_mko_ids":["mko-standard-normal-distribution"],"title_zh":"標準常態分布","stem_zh":"standard normal的Gaussian density、evenness與declared normalization責任。操作=normaldist-normalization-moment-boundary；資料={\"window\":25,\"panels\":34}。請依 exact_ordered_tuple_declared_by_interface 欄序回答。","answer":{"kind":"expression","value":"(25, 34, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)","explanation_zh":"Wave6第八層核對=(25, 34, finite_quadrature_only, 否, named_fact:integral_R_phi=1, named_fact:E[Z]=0, named_fact:Var(Z)=1, finite_window_or_panel_count_does_not_prove_normalization_or_moments)；依完整明示資料獨立重算；Phi(0)=1/2與Phi(-x)=1-Phi(x)只由已宣告normalization及symmetry推出；Gaussian integral、mean 0與variance 1是named theorem facts；finite quadrature不證它們或非零CDF exact value。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["標準常態分布"]},"variant":{"family_id":"qf-mod45-normaldist-normalization-moment-boundary","seed":202611144100,"parameters":{"window":25,"panels":34,"domain":"Symbolic_standard_normal_distribution","operation":"normaldist-normalization-moment-boundary","semantic_template":"qf-mod45-normaldist-normalization-moment-boundary/{\"panels\":34,\"window\":[25,1]}"}},"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"}}