{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0001","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-12,13],\"y_values\":[-3,20],\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-12, 13], [-3, 20], [1/2, 1/2], [1/2, 1/2], [[-12, -3, 1/2, 1/4], [-12, 20, 0, 1/4], [13, -3, 0, 1/4], [13, 20, 1/2, 1/4]], 否)","explanation_zh":"Wave4第二層核對=([-12, 13], [-3, 20], [1/2, 1/2], [1/2, 1/2], [[-12, -3, 1/2, 1/4], [-12, 20, 0, 1/4], [13, -3, 0, 1/4], [13, 20, 1/2, 1/4]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286001,"parameters":{"x_values":[-12,13],"y_values":[-3,20],"masses":[[[1,2],[0,1]],[[0,1],[1,2]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[-12,13],\"y_values\":[-3,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0002","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-11,14],\"y_values\":[-2,21],\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-11, 14], [-2, 21], [5/7, 2/7], [6/7, 1/7], [], 是)","explanation_zh":"Wave4第二層核對=([-11, 14], [-2, 21], [5/7, 2/7], [6/7, 1/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286002,"parameters":{"x_values":[-11,14],"y_values":[-2,21],"masses":[[[30,49],[5,49]],[[12,49],[2,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_values\":[-11,14],\"y_values\":[-2,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0003","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-10,15],\"y_values\":[-1,22],\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], 是)","explanation_zh":"Wave4第二層核對=([-10, 15], [-1, 22], [4/7, 3/7], [6/7, 1/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286003,"parameters":{"x_values":[-10,15],"y_values":[-1,22],"masses":[[[24,49],[4,49]],[[18,49],[3,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_values\":[-10,15],\"y_values\":[-1,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0004","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-9,16],\"y_values\":[0,23],\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-9, 16], [0, 23], [3/7, 4/7], [6/7, 1/7], [], 是)","explanation_zh":"Wave4第二層核對=([-9, 16], [0, 23], [3/7, 4/7], [6/7, 1/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286004,"parameters":{"x_values":[-9,16],"y_values":[0,23],"masses":[[[18,49],[3,49]],[[24,49],[4,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_values\":[-9,16],\"y_values\":[0,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0005","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-8,17],\"y_values\":[1,24],\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-8, 17], [1, 24], [2/7, 5/7], [6/7, 1/7], [], 是)","explanation_zh":"Wave4第二層核對=([-8, 17], [1, 24], [2/7, 5/7], [6/7, 1/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286005,"parameters":{"x_values":[-8,17],"y_values":[1,24],"masses":[[[12,49],[2,49]],[[30,49],[5,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_values\":[-8,17],\"y_values\":[1,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0006","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-7,18],\"y_values\":[2,25],\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-7, 18], [2, 25], [1/2, 1/2], [1/2, 1/2], [[-7, 2, 1/2, 1/4], [-7, 25, 0, 1/4], [18, 2, 0, 1/4], [18, 25, 1/2, 1/4]], 否)","explanation_zh":"Wave4第二層核對=([-7, 18], [2, 25], [1/2, 1/2], [1/2, 1/2], [[-7, 2, 1/2, 1/4], [-7, 25, 0, 1/4], [18, 2, 0, 1/4], [18, 25, 1/2, 1/4]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286006,"parameters":{"x_values":[-7,18],"y_values":[2,25],"masses":[[[1,2],[0,1]],[[0,1],[1,2]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[-7,18],\"y_values\":[2,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0007","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-6,19],\"y_values\":[3,26],\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-6, 19], [3, 26], [5/7, 2/7], [5/7, 2/7], [], 是)","explanation_zh":"Wave4第二層核對=([-6, 19], [3, 26], [5/7, 2/7], [5/7, 2/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286007,"parameters":{"x_values":[-6,19],"y_values":[3,26],"masses":[[[25,49],[10,49]],[[10,49],[4,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_values\":[-6,19],\"y_values\":[3,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0008","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-5,20],\"y_values\":[-3,27],\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-5, 20], [-3, 27], [4/7, 3/7], [5/7, 2/7], [], 是)","explanation_zh":"Wave4第二層核對=([-5, 20], [-3, 27], [4/7, 3/7], [5/7, 2/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286008,"parameters":{"x_values":[-5,20],"y_values":[-3,27],"masses":[[[20,49],[8,49]],[[15,49],[6,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_values\":[-5,20],\"y_values\":[-3,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0009","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-4,21],\"y_values\":[-2,28],\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-4, 21], [-2, 28], [3/7, 4/7], [5/7, 2/7], [], 是)","explanation_zh":"Wave4第二層核對=([-4, 21], [-2, 28], [3/7, 4/7], [5/7, 2/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286009,"parameters":{"x_values":[-4,21],"y_values":[-2,28],"masses":[[[15,49],[6,49]],[[20,49],[8,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_values\":[-4,21],\"y_values\":[-2,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0010","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-3,22],\"y_values\":[-1,29],\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-3, 22], [-1, 29], [2/7, 5/7], [5/7, 2/7], [], 是)","explanation_zh":"Wave4第二層核對=([-3, 22], [-1, 29], [2/7, 5/7], [5/7, 2/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286010,"parameters":{"x_values":[-3,22],"y_values":[-1,29],"masses":[[[10,49],[4,49]],[[25,49],[10,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_values\":[-3,22],\"y_values\":[-1,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0011","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-2,23],\"y_values\":[0,30],\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-2, 23], [0, 30], [1/2, 1/2], [1/2, 1/2], [[-2, 0, 1/2, 1/4], [-2, 30, 0, 1/4], [23, 0, 0, 1/4], [23, 30, 1/2, 1/4]], 否)","explanation_zh":"Wave4第二層核對=([-2, 23], [0, 30], [1/2, 1/2], [1/2, 1/2], [[-2, 0, 1/2, 1/4], [-2, 30, 0, 1/4], [23, 0, 0, 1/4], [23, 30, 1/2, 1/4]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286011,"parameters":{"x_values":[-2,23],"y_values":[0,30],"masses":[[[1,2],[0,1]],[[0,1],[1,2]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[-2,23],\"y_values\":[0,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0012","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[-1,24],\"y_values\":[1,31],\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([-1, 24], [1, 31], [5/7, 2/7], [4/7, 3/7], [], 是)","explanation_zh":"Wave4第二層核對=([-1, 24], [1, 31], [5/7, 2/7], [4/7, 3/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286012,"parameters":{"x_values":[-1,24],"y_values":[1,31],"masses":[[[20,49],[15,49]],[[8,49],[6,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_values\":[-1,24],\"y_values\":[1,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0013","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[0,25],\"y_values\":[2,32],\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([0, 25], [2, 32], [4/7, 3/7], [4/7, 3/7], [], 是)","explanation_zh":"Wave4第二層核對=([0, 25], [2, 32], [4/7, 3/7], [4/7, 3/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286013,"parameters":{"x_values":[0,25],"y_values":[2,32],"masses":[[[16,49],[12,49]],[[12,49],[9,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_values\":[0,25],\"y_values\":[2,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0014","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[1,26],\"y_values\":[3,33],\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([1, 26], [3, 33], [3/7, 4/7], [4/7, 3/7], [], 是)","explanation_zh":"Wave4第二層核對=([1, 26], [3, 33], [3/7, 4/7], [4/7, 3/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286014,"parameters":{"x_values":[1,26],"y_values":[3,33],"masses":[[[12,49],[9,49]],[[16,49],[12,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_values\":[1,26],\"y_values\":[3,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0015","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[2,27],\"y_values\":[-3,34],\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], 是)","explanation_zh":"Wave4第二層核對=([2, 27], [-3, 34], [2/7, 5/7], [4/7, 3/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286015,"parameters":{"x_values":[2,27],"y_values":[-3,34],"masses":[[[8,49],[6,49]],[[20,49],[15,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_values\":[2,27],\"y_values\":[-3,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0016","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[3,28],\"y_values\":[-2,35],\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([3, 28], [-2, 35], [1/2, 1/2], [1/2, 1/2], [[3, -2, 1/2, 1/4], [3, 35, 0, 1/4], [28, -2, 0, 1/4], [28, 35, 1/2, 1/4]], 否)","explanation_zh":"Wave4第二層核對=([3, 28], [-2, 35], [1/2, 1/2], [1/2, 1/2], [[3, -2, 1/2, 1/4], [3, 35, 0, 1/4], [28, -2, 0, 1/4], [28, 35, 1/2, 1/4]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286016,"parameters":{"x_values":[3,28],"y_values":[-2,35],"masses":[[[1,2],[0,1]],[[0,1],[1,2]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[3,28],\"y_values\":[-2,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0017","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[4,29],\"y_values\":[-1,36],\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([4, 29], [-1, 36], [5/7, 2/7], [3/7, 4/7], [], 是)","explanation_zh":"Wave4第二層核對=([4, 29], [-1, 36], [5/7, 2/7], [3/7, 4/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286017,"parameters":{"x_values":[4,29],"y_values":[-1,36],"masses":[[[15,49],[20,49]],[[6,49],[8,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_values\":[4,29],\"y_values\":[-1,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0018","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[5,30],\"y_values\":[0,37],\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([5, 30], [0, 37], [4/7, 3/7], [3/7, 4/7], [], 是)","explanation_zh":"Wave4第二層核對=([5, 30], [0, 37], [4/7, 3/7], [3/7, 4/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286018,"parameters":{"x_values":[5,30],"y_values":[0,37],"masses":[[[12,49],[16,49]],[[9,49],[12,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_values\":[5,30],\"y_values\":[0,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0019","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[6,31],\"y_values\":[1,38],\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([6, 31], [1, 38], [3/7, 4/7], [3/7, 4/7], [], 是)","explanation_zh":"Wave4第二層核對=([6, 31], [1, 38], [3/7, 4/7], [3/7, 4/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286019,"parameters":{"x_values":[6,31],"y_values":[1,38],"masses":[[[9,49],[12,49]],[[12,49],[16,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_values\":[6,31],\"y_values\":[1,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0020","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[7,32],\"y_values\":[2,39],\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([7, 32], [2, 39], [2/7, 5/7], [3/7, 4/7], [], 是)","explanation_zh":"Wave4第二層核對=([7, 32], [2, 39], [2/7, 5/7], [3/7, 4/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286020,"parameters":{"x_values":[7,32],"y_values":[2,39],"masses":[[[6,49],[8,49]],[[15,49],[20,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_values\":[7,32],\"y_values\":[2,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0021","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[8,33],\"y_values\":[3,40],\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([8, 33], [3, 40], [1/2, 1/2], [1/2, 1/2], [[8, 3, 1/2, 1/4], [8, 40, 0, 1/4], [33, 3, 0, 1/4], [33, 40, 1/2, 1/4]], 否)","explanation_zh":"Wave4第二層核對=([8, 33], [3, 40], [1/2, 1/2], [1/2, 1/2], [[8, 3, 1/2, 1/4], [8, 40, 0, 1/4], [33, 3, 0, 1/4], [33, 40, 1/2, 1/4]], 否)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286021,"parameters":{"x_values":[8,33],"y_values":[3,40],"masses":[[[1,2],[0,1]],[[0,1],[1,2]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[1,2],[0,1]],[[0,1],[1,2]]],\"x_values\":[8,33],\"y_values\":[3,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0022","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[9,34],\"y_values\":[-3,41],\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([9, 34], [-3, 41], [5/7, 2/7], [2/7, 5/7], [], 是)","explanation_zh":"Wave4第二層核對=([9, 34], [-3, 41], [5/7, 2/7], [2/7, 5/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286022,"parameters":{"x_values":[9,34],"y_values":[-3,41],"masses":[[[10,49],[25,49]],[[4,49],[10,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_values\":[9,34],\"y_values\":[-3,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0023","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[10,35],\"y_values\":[-2,42],\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([10, 35], [-2, 42], [4/7, 3/7], [2/7, 5/7], [], 是)","explanation_zh":"Wave4第二層核對=([10, 35], [-2, 42], [4/7, 3/7], [2/7, 5/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286023,"parameters":{"x_values":[10,35],"y_values":[-2,42],"masses":[[[8,49],[20,49]],[[6,49],[15,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_values\":[10,35],\"y_values\":[-2,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0024","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[11,36],\"y_values\":[-1,43],\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([11, 36], [-1, 43], [3/7, 4/7], [2/7, 5/7], [], 是)","explanation_zh":"Wave4第二層核對=([11, 36], [-1, 43], [3/7, 4/7], [2/7, 5/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286024,"parameters":{"x_values":[11,36],"y_values":[-1,43],"masses":[[[6,49],[15,49]],[[8,49],[20,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_values\":[11,36],\"y_values\":[-1,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0025","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-qualification；資料={\"x_values\":[12,37],\"y_values\":[0,44],\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]]}。請依 [x_values,y_values,x_marginal,y_marginal,factorization_failures,independent] 欄序回答。","answer":{"kind":"expression","value":"([12, 37], [0, 44], [2/7, 5/7], [2/7, 5/7], [], 是)","explanation_zh":"Wave4第二層核對=([12, 37], [0, 44], [2/7, 5/7], [2/7, 5/7], [], 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-qualification","seed":202610286025,"parameters":{"x_values":[12,37],"y_values":[0,44],"masses":[[[4,49],[10,49]],[[10,49],[25,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-qualification","semantic_template":"qf-mod28-rvind-qualification/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_values\":[12,37],\"y_values\":[0,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0026","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-12,13],\"y_values\":[-3,20],\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x\":-12,\"y\":20}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(-12, 20, 6/49, 6/7, 1/7, 6/49, 是)","explanation_zh":"Wave4第二層核對=(-12, 20, 6/49, 6/7, 1/7, 6/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286026,"parameters":{"x_values":[-12,13],"y_values":[-3,20],"masses":[[[36,49],[6,49]],[[6,49],[1,49]]],"x":-12,"y":20,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x\":-12,\"x_values\":[-12,13],\"y\":20,\"y_values\":[-3,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0027","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-11,14],\"y_values\":[-2,21],\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x\":14,\"y\":-2}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(14, -2, 12/49, 2/7, 6/7, 12/49, 是)","explanation_zh":"Wave4第二層核對=(14, -2, 12/49, 2/7, 6/7, 12/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286027,"parameters":{"x_values":[-11,14],"y_values":[-2,21],"masses":[[[30,49],[5,49]],[[12,49],[2,49]]],"x":14,"y":-2,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x\":14,\"x_values\":[-11,14],\"y\":-2,\"y_values\":[-2,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0028","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-10,15],\"y_values\":[-1,22],\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x\":-10,\"y\":22}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(-10, 22, 4/49, 4/7, 1/7, 4/49, 是)","explanation_zh":"Wave4第二層核對=(-10, 22, 4/49, 4/7, 1/7, 4/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286028,"parameters":{"x_values":[-10,15],"y_values":[-1,22],"masses":[[[24,49],[4,49]],[[18,49],[3,49]]],"x":-10,"y":22,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x\":-10,\"x_values\":[-10,15],\"y\":22,\"y_values\":[-1,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0029","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-9,16],\"y_values\":[0,23],\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x\":16,\"y\":0}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(16, 0, 24/49, 4/7, 6/7, 24/49, 是)","explanation_zh":"Wave4第二層核對=(16, 0, 24/49, 4/7, 6/7, 24/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286029,"parameters":{"x_values":[-9,16],"y_values":[0,23],"masses":[[[18,49],[3,49]],[[24,49],[4,49]]],"x":16,"y":0,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x\":16,\"x_values\":[-9,16],\"y\":0,\"y_values\":[0,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0030","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-8,17],\"y_values\":[1,24],\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x\":-8,\"y\":24}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(-8, 24, 2/49, 2/7, 1/7, 2/49, 是)","explanation_zh":"Wave4第二層核對=(-8, 24, 2/49, 2/7, 1/7, 2/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286030,"parameters":{"x_values":[-8,17],"y_values":[1,24],"masses":[[[12,49],[2,49]],[[30,49],[5,49]]],"x":-8,"y":24,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x\":-8,\"x_values\":[-8,17],\"y\":24,\"y_values\":[1,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0031","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-7,18],\"y_values\":[2,25],\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x\":18,\"y\":2}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(18, 2, 5/49, 1/7, 5/7, 5/49, 是)","explanation_zh":"Wave4第二層核對=(18, 2, 5/49, 1/7, 5/7, 5/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286031,"parameters":{"x_values":[-7,18],"y_values":[2,25],"masses":[[[30,49],[12,49]],[[5,49],[2,49]]],"x":18,"y":2,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x\":18,\"x_values\":[-7,18],\"y\":2,\"y_values\":[2,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0032","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-6,19],\"y_values\":[3,26],\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x\":-6,\"y\":26}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(-6, 26, 10/49, 5/7, 2/7, 10/49, 是)","explanation_zh":"Wave4第二層核對=(-6, 26, 10/49, 5/7, 2/7, 10/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286032,"parameters":{"x_values":[-6,19],"y_values":[3,26],"masses":[[[25,49],[10,49]],[[10,49],[4,49]]],"x":-6,"y":26,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x\":-6,\"x_values\":[-6,19],\"y\":26,\"y_values\":[3,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0033","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-5,20],\"y_values\":[-3,27],\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x\":20,\"y\":-3}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(20, -3, 15/49, 3/7, 5/7, 15/49, 是)","explanation_zh":"Wave4第二層核對=(20, -3, 15/49, 3/7, 5/7, 15/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286033,"parameters":{"x_values":[-5,20],"y_values":[-3,27],"masses":[[[20,49],[8,49]],[[15,49],[6,49]]],"x":20,"y":-3,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x\":20,\"x_values\":[-5,20],\"y\":-3,\"y_values\":[-3,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0034","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-4,21],\"y_values\":[-2,28],\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x\":-4,\"y\":28}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(-4, 28, 6/49, 3/7, 2/7, 6/49, 是)","explanation_zh":"Wave4第二層核對=(-4, 28, 6/49, 3/7, 2/7, 6/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286034,"parameters":{"x_values":[-4,21],"y_values":[-2,28],"masses":[[[15,49],[6,49]],[[20,49],[8,49]]],"x":-4,"y":28,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x\":-4,\"x_values\":[-4,21],\"y\":28,\"y_values\":[-2,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0035","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-3,22],\"y_values\":[-1,29],\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x\":22,\"y\":-1}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(22, -1, 25/49, 5/7, 5/7, 25/49, 是)","explanation_zh":"Wave4第二層核對=(22, -1, 25/49, 5/7, 5/7, 25/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286035,"parameters":{"x_values":[-3,22],"y_values":[-1,29],"masses":[[[10,49],[4,49]],[[25,49],[10,49]]],"x":22,"y":-1,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x\":22,\"x_values\":[-3,22],\"y\":-1,\"y_values\":[-1,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0036","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-2,23],\"y_values\":[0,30],\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x\":-2,\"y\":30}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(-2, 30, 18/49, 6/7, 3/7, 18/49, 是)","explanation_zh":"Wave4第二層核對=(-2, 30, 18/49, 6/7, 3/7, 18/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286036,"parameters":{"x_values":[-2,23],"y_values":[0,30],"masses":[[[24,49],[18,49]],[[4,49],[3,49]]],"x":-2,"y":30,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x\":-2,\"x_values\":[-2,23],\"y\":30,\"y_values\":[0,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0037","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[-1,24],\"y_values\":[1,31],\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x\":24,\"y\":1}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(24, 1, 8/49, 2/7, 4/7, 8/49, 是)","explanation_zh":"Wave4第二層核對=(24, 1, 8/49, 2/7, 4/7, 8/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286037,"parameters":{"x_values":[-1,24],"y_values":[1,31],"masses":[[[20,49],[15,49]],[[8,49],[6,49]]],"x":24,"y":1,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x\":24,\"x_values\":[-1,24],\"y\":1,\"y_values\":[1,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0038","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[0,25],\"y_values\":[2,32],\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x\":0,\"y\":32}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(0, 32, 12/49, 4/7, 3/7, 12/49, 是)","explanation_zh":"Wave4第二層核對=(0, 32, 12/49, 4/7, 3/7, 12/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286038,"parameters":{"x_values":[0,25],"y_values":[2,32],"masses":[[[16,49],[12,49]],[[12,49],[9,49]]],"x":0,"y":32,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x\":0,\"x_values\":[0,25],\"y\":32,\"y_values\":[2,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0039","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[1,26],\"y_values\":[3,33],\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x\":26,\"y\":3}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(26, 3, 16/49, 4/7, 4/7, 16/49, 是)","explanation_zh":"Wave4第二層核對=(26, 3, 16/49, 4/7, 4/7, 16/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286039,"parameters":{"x_values":[1,26],"y_values":[3,33],"masses":[[[12,49],[9,49]],[[16,49],[12,49]]],"x":26,"y":3,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x\":26,\"x_values\":[1,26],\"y\":3,\"y_values\":[3,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0040","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[2,27],\"y_values\":[-3,34],\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x\":2,\"y\":34}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(2, 34, 6/49, 2/7, 3/7, 6/49, 是)","explanation_zh":"Wave4第二層核對=(2, 34, 6/49, 2/7, 3/7, 6/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286040,"parameters":{"x_values":[2,27],"y_values":[-3,34],"masses":[[[8,49],[6,49]],[[20,49],[15,49]]],"x":2,"y":34,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x\":2,\"x_values\":[2,27],\"y\":34,\"y_values\":[-3,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0041","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[3,28],\"y_values\":[-2,35],\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x\":28,\"y\":-2}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(28, -2, 3/49, 1/7, 3/7, 3/49, 是)","explanation_zh":"Wave4第二層核對=(28, -2, 3/49, 1/7, 3/7, 3/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286041,"parameters":{"x_values":[3,28],"y_values":[-2,35],"masses":[[[18,49],[24,49]],[[3,49],[4,49]]],"x":28,"y":-2,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x\":28,\"x_values\":[3,28],\"y\":-2,\"y_values\":[-2,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0042","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[4,29],\"y_values\":[-1,36],\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x\":4,\"y\":36}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(4, 36, 20/49, 5/7, 4/7, 20/49, 是)","explanation_zh":"Wave4第二層核對=(4, 36, 20/49, 5/7, 4/7, 20/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286042,"parameters":{"x_values":[4,29],"y_values":[-1,36],"masses":[[[15,49],[20,49]],[[6,49],[8,49]]],"x":4,"y":36,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x\":4,\"x_values\":[4,29],\"y\":36,\"y_values\":[-1,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0043","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[5,30],\"y_values\":[0,37],\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x\":30,\"y\":0}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(30, 0, 9/49, 3/7, 3/7, 9/49, 是)","explanation_zh":"Wave4第二層核對=(30, 0, 9/49, 3/7, 3/7, 9/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286043,"parameters":{"x_values":[5,30],"y_values":[0,37],"masses":[[[12,49],[16,49]],[[9,49],[12,49]]],"x":30,"y":0,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x\":30,\"x_values\":[5,30],\"y\":0,\"y_values\":[0,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0044","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[6,31],\"y_values\":[1,38],\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x\":6,\"y\":38}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(6, 38, 12/49, 3/7, 4/7, 12/49, 是)","explanation_zh":"Wave4第二層核對=(6, 38, 12/49, 3/7, 4/7, 12/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286044,"parameters":{"x_values":[6,31],"y_values":[1,38],"masses":[[[9,49],[12,49]],[[12,49],[16,49]]],"x":6,"y":38,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x\":6,\"x_values\":[6,31],\"y\":38,\"y_values\":[1,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0045","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[7,32],\"y_values\":[2,39],\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x\":32,\"y\":2}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(32, 2, 15/49, 5/7, 3/7, 15/49, 是)","explanation_zh":"Wave4第二層核對=(32, 2, 15/49, 5/7, 3/7, 15/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286045,"parameters":{"x_values":[7,32],"y_values":[2,39],"masses":[[[6,49],[8,49]],[[15,49],[20,49]]],"x":32,"y":2,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x\":32,\"x_values\":[7,32],\"y\":2,\"y_values\":[2,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0046","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[8,33],\"y_values\":[3,40],\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x\":8,\"y\":40}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(8, 40, 30/49, 6/7, 5/7, 30/49, 是)","explanation_zh":"Wave4第二層核對=(8, 40, 30/49, 6/7, 5/7, 30/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286046,"parameters":{"x_values":[8,33],"y_values":[3,40],"masses":[[[12,49],[30,49]],[[2,49],[5,49]]],"x":8,"y":40,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x\":8,\"x_values\":[8,33],\"y\":40,\"y_values\":[3,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0047","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[9,34],\"y_values\":[-3,41],\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x\":34,\"y\":-3}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(34, -3, 4/49, 2/7, 2/7, 4/49, 是)","explanation_zh":"Wave4第二層核對=(34, -3, 4/49, 2/7, 2/7, 4/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286047,"parameters":{"x_values":[9,34],"y_values":[-3,41],"masses":[[[10,49],[25,49]],[[4,49],[10,49]]],"x":34,"y":-3,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x\":34,\"x_values\":[9,34],\"y\":-3,\"y_values\":[-3,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0048","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[10,35],\"y_values\":[-2,42],\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x\":10,\"y\":42}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(10, 42, 20/49, 4/7, 5/7, 20/49, 是)","explanation_zh":"Wave4第二層核對=(10, 42, 20/49, 4/7, 5/7, 20/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286048,"parameters":{"x_values":[10,35],"y_values":[-2,42],"masses":[[[8,49],[20,49]],[[6,49],[15,49]]],"x":10,"y":42,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x\":10,\"x_values\":[10,35],\"y\":42,\"y_values\":[-2,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0049","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[11,36],\"y_values\":[-1,43],\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x\":36,\"y\":-1}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(36, -1, 8/49, 4/7, 2/7, 8/49, 是)","explanation_zh":"Wave4第二層核對=(36, -1, 8/49, 4/7, 2/7, 8/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286049,"parameters":{"x_values":[11,36],"y_values":[-1,43],"masses":[[[6,49],[15,49]],[[8,49],[20,49]]],"x":36,"y":-1,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x\":36,\"x_values\":[11,36],\"y\":-1,\"y_values\":[-1,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0050","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-marginal-product；資料={\"x_values\":[12,37],\"y_values\":[0,44],\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x\":12,\"y\":44}。請依 [x,y,joint_mass,x_mass,y_mass,product,equal] 欄序回答。","answer":{"kind":"expression","value":"(12, 44, 10/49, 2/7, 5/7, 10/49, 是)","explanation_zh":"Wave4第二層核對=(12, 44, 10/49, 2/7, 5/7, 10/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-marginal-product","seed":202610286050,"parameters":{"x_values":[12,37],"y_values":[0,44],"masses":[[[4,49],[10,49]],[[10,49],[25,49]]],"x":12,"y":44,"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-marginal-product","semantic_template":"qf-mod28-rvind-marginal-product/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x\":12,\"x_values\":[12,37],\"y\":44,\"y_values\":[0,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0051","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-12,13],\"y_values\":[-3,20],\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x_event\":[-12],\"y_event\":[20]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([-12], [20], 6/7, 1/7, 6/49, 6/49, 是)","explanation_zh":"Wave4第二層核對=([-12], [20], 6/7, 1/7, 6/49, 6/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286051,"parameters":{"x_values":[-12,13],"y_values":[-3,20],"masses":[[[36,49],[6,49]],[[6,49],[1,49]]],"x_event":[-12],"y_event":[20],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x_event\":[-12],\"x_values\":[-12,13],\"y_event\":[20],\"y_values\":[-3,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0052","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-11,14],\"y_values\":[-2,21],\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_event\":[14],\"y_event\":[-2]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([14], [-2], 2/7, 6/7, 12/49, 12/49, 是)","explanation_zh":"Wave4第二層核對=([14], [-2], 2/7, 6/7, 12/49, 12/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286052,"parameters":{"x_values":[-11,14],"y_values":[-2,21],"masses":[[[30,49],[5,49]],[[12,49],[2,49]]],"x_event":[14],"y_event":[-2],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_event\":[14],\"x_values\":[-11,14],\"y_event\":[-2],\"y_values\":[-2,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0053","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-10,15],\"y_values\":[-1,22],\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_event\":[-10],\"y_event\":[22]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([-10], [22], 4/7, 1/7, 4/49, 4/49, 是)","explanation_zh":"Wave4第二層核對=([-10], [22], 4/7, 1/7, 4/49, 4/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286053,"parameters":{"x_values":[-10,15],"y_values":[-1,22],"masses":[[[24,49],[4,49]],[[18,49],[3,49]]],"x_event":[-10],"y_event":[22],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_event\":[-10],\"x_values\":[-10,15],\"y_event\":[22],\"y_values\":[-1,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0054","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-9,16],\"y_values\":[0,23],\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_event\":[16],\"y_event\":[0]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([16], [0], 4/7, 6/7, 24/49, 24/49, 是)","explanation_zh":"Wave4第二層核對=([16], [0], 4/7, 6/7, 24/49, 24/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286054,"parameters":{"x_values":[-9,16],"y_values":[0,23],"masses":[[[18,49],[3,49]],[[24,49],[4,49]]],"x_event":[16],"y_event":[0],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_event\":[16],\"x_values\":[-9,16],\"y_event\":[0],\"y_values\":[0,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0055","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-8,17],\"y_values\":[1,24],\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_event\":[-8],\"y_event\":[24]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([-8], [24], 2/7, 1/7, 2/49, 2/49, 是)","explanation_zh":"Wave4第二層核對=([-8], [24], 2/7, 1/7, 2/49, 2/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286055,"parameters":{"x_values":[-8,17],"y_values":[1,24],"masses":[[[12,49],[2,49]],[[30,49],[5,49]]],"x_event":[-8],"y_event":[24],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_event\":[-8],\"x_values\":[-8,17],\"y_event\":[24],\"y_values\":[1,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0056","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-7,18],\"y_values\":[2,25],\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x_event\":[18],\"y_event\":[2]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([18], [2], 1/7, 5/7, 5/49, 5/49, 是)","explanation_zh":"Wave4第二層核對=([18], [2], 1/7, 5/7, 5/49, 5/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286056,"parameters":{"x_values":[-7,18],"y_values":[2,25],"masses":[[[30,49],[12,49]],[[5,49],[2,49]]],"x_event":[18],"y_event":[2],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x_event\":[18],\"x_values\":[-7,18],\"y_event\":[2],\"y_values\":[2,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0057","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-6,19],\"y_values\":[3,26],\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_event\":[-6],\"y_event\":[26]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([-6], [26], 5/7, 2/7, 10/49, 10/49, 是)","explanation_zh":"Wave4第二層核對=([-6], [26], 5/7, 2/7, 10/49, 10/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286057,"parameters":{"x_values":[-6,19],"y_values":[3,26],"masses":[[[25,49],[10,49]],[[10,49],[4,49]]],"x_event":[-6],"y_event":[26],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_event\":[-6],\"x_values\":[-6,19],\"y_event\":[26],\"y_values\":[3,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0058","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-5,20],\"y_values\":[-3,27],\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_event\":[20],\"y_event\":[-3]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([20], [-3], 3/7, 5/7, 15/49, 15/49, 是)","explanation_zh":"Wave4第二層核對=([20], [-3], 3/7, 5/7, 15/49, 15/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286058,"parameters":{"x_values":[-5,20],"y_values":[-3,27],"masses":[[[20,49],[8,49]],[[15,49],[6,49]]],"x_event":[20],"y_event":[-3],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_event\":[20],\"x_values\":[-5,20],\"y_event\":[-3],\"y_values\":[-3,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0059","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-4,21],\"y_values\":[-2,28],\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_event\":[-4],\"y_event\":[28]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([-4], [28], 3/7, 2/7, 6/49, 6/49, 是)","explanation_zh":"Wave4第二層核對=([-4], [28], 3/7, 2/7, 6/49, 6/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286059,"parameters":{"x_values":[-4,21],"y_values":[-2,28],"masses":[[[15,49],[6,49]],[[20,49],[8,49]]],"x_event":[-4],"y_event":[28],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_event\":[-4],\"x_values\":[-4,21],\"y_event\":[28],\"y_values\":[-2,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0060","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-3,22],\"y_values\":[-1,29],\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_event\":[22],\"y_event\":[-1]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([22], [-1], 5/7, 5/7, 25/49, 25/49, 是)","explanation_zh":"Wave4第二層核對=([22], [-1], 5/7, 5/7, 25/49, 25/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286060,"parameters":{"x_values":[-3,22],"y_values":[-1,29],"masses":[[[10,49],[4,49]],[[25,49],[10,49]]],"x_event":[22],"y_event":[-1],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_event\":[22],\"x_values\":[-3,22],\"y_event\":[-1],\"y_values\":[-1,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0061","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-2,23],\"y_values\":[0,30],\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x_event\":[-2],\"y_event\":[30]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([-2], [30], 6/7, 3/7, 18/49, 18/49, 是)","explanation_zh":"Wave4第二層核對=([-2], [30], 6/7, 3/7, 18/49, 18/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286061,"parameters":{"x_values":[-2,23],"y_values":[0,30],"masses":[[[24,49],[18,49]],[[4,49],[3,49]]],"x_event":[-2],"y_event":[30],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x_event\":[-2],\"x_values\":[-2,23],\"y_event\":[30],\"y_values\":[0,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0062","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[-1,24],\"y_values\":[1,31],\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_event\":[24],\"y_event\":[1]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([24], [1], 2/7, 4/7, 8/49, 8/49, 是)","explanation_zh":"Wave4第二層核對=([24], [1], 2/7, 4/7, 8/49, 8/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286062,"parameters":{"x_values":[-1,24],"y_values":[1,31],"masses":[[[20,49],[15,49]],[[8,49],[6,49]]],"x_event":[24],"y_event":[1],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_event\":[24],\"x_values\":[-1,24],\"y_event\":[1],\"y_values\":[1,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0063","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[0,25],\"y_values\":[2,32],\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_event\":[0],\"y_event\":[32]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([0], [32], 4/7, 3/7, 12/49, 12/49, 是)","explanation_zh":"Wave4第二層核對=([0], [32], 4/7, 3/7, 12/49, 12/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286063,"parameters":{"x_values":[0,25],"y_values":[2,32],"masses":[[[16,49],[12,49]],[[12,49],[9,49]]],"x_event":[0],"y_event":[32],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_event\":[0],\"x_values\":[0,25],\"y_event\":[32],\"y_values\":[2,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0064","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[1,26],\"y_values\":[3,33],\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_event\":[26],\"y_event\":[3]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([26], [3], 4/7, 4/7, 16/49, 16/49, 是)","explanation_zh":"Wave4第二層核對=([26], [3], 4/7, 4/7, 16/49, 16/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286064,"parameters":{"x_values":[1,26],"y_values":[3,33],"masses":[[[12,49],[9,49]],[[16,49],[12,49]]],"x_event":[26],"y_event":[3],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_event\":[26],\"x_values\":[1,26],\"y_event\":[3],\"y_values\":[3,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0065","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[2,27],\"y_values\":[-3,34],\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_event\":[2],\"y_event\":[34]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([2], [34], 2/7, 3/7, 6/49, 6/49, 是)","explanation_zh":"Wave4第二層核對=([2], [34], 2/7, 3/7, 6/49, 6/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286065,"parameters":{"x_values":[2,27],"y_values":[-3,34],"masses":[[[8,49],[6,49]],[[20,49],[15,49]]],"x_event":[2],"y_event":[34],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_event\":[2],\"x_values\":[2,27],\"y_event\":[34],\"y_values\":[-3,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0066","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[3,28],\"y_values\":[-2,35],\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x_event\":[28],\"y_event\":[-2]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([28], [-2], 1/7, 3/7, 3/49, 3/49, 是)","explanation_zh":"Wave4第二層核對=([28], [-2], 1/7, 3/7, 3/49, 3/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286066,"parameters":{"x_values":[3,28],"y_values":[-2,35],"masses":[[[18,49],[24,49]],[[3,49],[4,49]]],"x_event":[28],"y_event":[-2],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x_event\":[28],\"x_values\":[3,28],\"y_event\":[-2],\"y_values\":[-2,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0067","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[4,29],\"y_values\":[-1,36],\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_event\":[4],\"y_event\":[36]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([4], [36], 5/7, 4/7, 20/49, 20/49, 是)","explanation_zh":"Wave4第二層核對=([4], [36], 5/7, 4/7, 20/49, 20/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286067,"parameters":{"x_values":[4,29],"y_values":[-1,36],"masses":[[[15,49],[20,49]],[[6,49],[8,49]]],"x_event":[4],"y_event":[36],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_event\":[4],\"x_values\":[4,29],\"y_event\":[36],\"y_values\":[-1,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0068","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[5,30],\"y_values\":[0,37],\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_event\":[30],\"y_event\":[0]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([30], [0], 3/7, 3/7, 9/49, 9/49, 是)","explanation_zh":"Wave4第二層核對=([30], [0], 3/7, 3/7, 9/49, 9/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286068,"parameters":{"x_values":[5,30],"y_values":[0,37],"masses":[[[12,49],[16,49]],[[9,49],[12,49]]],"x_event":[30],"y_event":[0],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_event\":[30],\"x_values\":[5,30],\"y_event\":[0],\"y_values\":[0,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0069","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[6,31],\"y_values\":[1,38],\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_event\":[6],\"y_event\":[38]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([6], [38], 3/7, 4/7, 12/49, 12/49, 是)","explanation_zh":"Wave4第二層核對=([6], [38], 3/7, 4/7, 12/49, 12/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286069,"parameters":{"x_values":[6,31],"y_values":[1,38],"masses":[[[9,49],[12,49]],[[12,49],[16,49]]],"x_event":[6],"y_event":[38],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_event\":[6],\"x_values\":[6,31],\"y_event\":[38],\"y_values\":[1,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0070","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[7,32],\"y_values\":[2,39],\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_event\":[32],\"y_event\":[2]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([32], [2], 5/7, 3/7, 15/49, 15/49, 是)","explanation_zh":"Wave4第二層核對=([32], [2], 5/7, 3/7, 15/49, 15/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286070,"parameters":{"x_values":[7,32],"y_values":[2,39],"masses":[[[6,49],[8,49]],[[15,49],[20,49]]],"x_event":[32],"y_event":[2],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_event\":[32],\"x_values\":[7,32],\"y_event\":[2],\"y_values\":[2,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0071","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[8,33],\"y_values\":[3,40],\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x_event\":[8],\"y_event\":[40]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([8], [40], 6/7, 5/7, 30/49, 30/49, 是)","explanation_zh":"Wave4第二層核對=([8], [40], 6/7, 5/7, 30/49, 30/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286071,"parameters":{"x_values":[8,33],"y_values":[3,40],"masses":[[[12,49],[30,49]],[[2,49],[5,49]]],"x_event":[8],"y_event":[40],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x_event\":[8],\"x_values\":[8,33],\"y_event\":[40],\"y_values\":[3,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0072","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[9,34],\"y_values\":[-3,41],\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"y_event\":[-3]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([34], [-3], 2/7, 2/7, 4/49, 4/49, 是)","explanation_zh":"Wave4第二層核對=([34], [-3], 2/7, 2/7, 4/49, 4/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286072,"parameters":{"x_values":[9,34],"y_values":[-3,41],"masses":[[[10,49],[25,49]],[[4,49],[10,49]]],"x_event":[34],"y_event":[-3],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_event\":[34],\"x_values\":[9,34],\"y_event\":[-3],\"y_values\":[-3,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0073","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[10,35],\"y_values\":[-2,42],\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_event\":[10],\"y_event\":[42]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([10], [42], 4/7, 5/7, 20/49, 20/49, 是)","explanation_zh":"Wave4第二層核對=([10], [42], 4/7, 5/7, 20/49, 20/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286073,"parameters":{"x_values":[10,35],"y_values":[-2,42],"masses":[[[8,49],[20,49]],[[6,49],[15,49]]],"x_event":[10],"y_event":[42],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_event\":[10],\"x_values\":[10,35],\"y_event\":[42],\"y_values\":[-2,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0074","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[11,36],\"y_values\":[-1,43],\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_event\":[36],\"y_event\":[-1]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([36], [-1], 4/7, 2/7, 8/49, 8/49, 是)","explanation_zh":"Wave4第二層核對=([36], [-1], 4/7, 2/7, 8/49, 8/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286074,"parameters":{"x_values":[11,36],"y_values":[-1,43],"masses":[[[6,49],[15,49]],[[8,49],[20,49]]],"x_event":[36],"y_event":[-1],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_event\":[36],\"x_values\":[11,36],\"y_event\":[-1],\"y_values\":[-1,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0075","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-event-preimage；資料={\"x_values\":[12,37],\"y_values\":[0,44],\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_event\":[12],\"y_event\":[44]}。請依 [x_event,y_event,P_X_event,P_Y_event,P_joint,product,equal] 欄序回答。","answer":{"kind":"expression","value":"([12], [44], 2/7, 5/7, 10/49, 10/49, 是)","explanation_zh":"Wave4第二層核對=([12], [44], 2/7, 5/7, 10/49, 10/49, 是)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-event-preimage","seed":202610286075,"parameters":{"x_values":[12,37],"y_values":[0,44],"masses":[[[4,49],[10,49]],[[10,49],[25,49]]],"x_event":[12],"y_event":[44],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-event-preimage","semantic_template":"qf-mod28-rvind-event-preimage/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_event\":[12],\"x_values\":[12,37],\"y_event\":[44],\"y_values\":[0,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0076","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-12,13],\"y_values\":[-3,20],\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, -59/7, 2/7, -118/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, -59/7, 2/7, -118/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286076,"parameters":{"x_values":[-12,13],"y_values":[-3,20],"masses":[[[36,49],[6,49]],[[6,49],[1,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[36,49],[6,49]],[[6,49],[1,49]]],\"x_values\":[-12,13],\"y_values\":[-3,20]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0077","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-11,14],\"y_values\":[-2,21],\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, -27/7, 9/7, -243/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, -27/7, 9/7, -243/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286077,"parameters":{"x_values":[-11,14],"y_values":[-2,21],"masses":[[[30,49],[5,49]],[[12,49],[2,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[30,49],[5,49]],[[12,49],[2,49]]],\"x_values\":[-11,14],\"y_values\":[-2,21]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0078","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-10,15],\"y_values\":[-1,22],\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 5/7, 16/7, 80/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 5/7, 16/7, 80/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286078,"parameters":{"x_values":[-10,15],"y_values":[-1,22],"masses":[[[24,49],[4,49]],[[18,49],[3,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[24,49],[4,49]],[[18,49],[3,49]]],\"x_values\":[-10,15],\"y_values\":[-1,22]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0079","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-9,16],\"y_values\":[0,23],\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 37/7, 23/7, 851/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 37/7, 23/7, 851/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286079,"parameters":{"x_values":[-9,16],"y_values":[0,23],"masses":[[[18,49],[3,49]],[[24,49],[4,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[18,49],[3,49]],[[24,49],[4,49]]],\"x_values\":[-9,16],\"y_values\":[0,23]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0080","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-8,17],\"y_values\":[1,24],\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 69/7, 30/7, 2070/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 69/7, 30/7, 2070/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286080,"parameters":{"x_values":[-8,17],"y_values":[1,24],"masses":[[[12,49],[2,49]],[[30,49],[5,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[2,49]],[[30,49],[5,49]]],\"x_values\":[-8,17],\"y_values\":[1,24]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0081","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-7,18],\"y_values\":[2,25],\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, -24/7, 60/7, -1440/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, -24/7, 60/7, -1440/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286081,"parameters":{"x_values":[-7,18],"y_values":[2,25],"masses":[[[30,49],[12,49]],[[5,49],[2,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[30,49],[12,49]],[[5,49],[2,49]]],\"x_values\":[-7,18],\"y_values\":[2,25]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0082","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-6,19],\"y_values\":[3,26],\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 8/7, 67/7, 536/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 8/7, 67/7, 536/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286082,"parameters":{"x_values":[-6,19],"y_values":[3,26],"masses":[[[25,49],[10,49]],[[10,49],[4,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[25,49],[10,49]],[[10,49],[4,49]]],\"x_values\":[-6,19],\"y_values\":[3,26]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0083","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-5,20],\"y_values\":[-3,27],\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 40/7, 39/7, 1560/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 40/7, 39/7, 1560/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":1,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286083,"parameters":{"x_values":[-5,20],"y_values":[-3,27],"masses":[[[20,49],[8,49]],[[15,49],[6,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[20,49],[8,49]],[[15,49],[6,49]]],\"x_values\":[-5,20],\"y_values\":[-3,27]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0084","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-4,21],\"y_values\":[-2,28],\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 72/7, 46/7, 3312/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 72/7, 46/7, 3312/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286084,"parameters":{"x_values":[-4,21],"y_values":[-2,28],"masses":[[[15,49],[6,49]],[[20,49],[8,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[15,49],[6,49]],[[20,49],[8,49]]],\"x_values\":[-4,21],\"y_values\":[-2,28]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0085","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-3,22],\"y_values\":[-1,29],\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 104/7, 53/7, 5512/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 104/7, 53/7, 5512/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286085,"parameters":{"x_values":[-3,22],"y_values":[-1,29],"masses":[[[10,49],[4,49]],[[25,49],[10,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[10,49],[4,49]],[[25,49],[10,49]]],\"x_values\":[-3,22],\"y_values\":[-1,29]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0086","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-2,23],\"y_values\":[0,30],\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 11/7, 90/7, 990/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 11/7, 90/7, 990/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286086,"parameters":{"x_values":[-2,23],"y_values":[0,30],"masses":[[[24,49],[18,49]],[[4,49],[3,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[24,49],[18,49]],[[4,49],[3,49]]],\"x_values\":[-2,23],\"y_values\":[0,30]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0087","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[-1,24],\"y_values\":[1,31],\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 43/7, 97/7, 4171/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 43/7, 97/7, 4171/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286087,"parameters":{"x_values":[-1,24],"y_values":[1,31],"masses":[[[20,49],[15,49]],[[8,49],[6,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[20,49],[15,49]],[[8,49],[6,49]]],\"x_values\":[-1,24],\"y_values\":[1,31]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0088","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[0,25],\"y_values\":[2,32],\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 75/7, 104/7, 7800/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 75/7, 104/7, 7800/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286088,"parameters":{"x_values":[0,25],"y_values":[2,32],"masses":[[[16,49],[12,49]],[[12,49],[9,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[16,49],[12,49]],[[12,49],[9,49]]],\"x_values\":[0,25],\"y_values\":[2,32]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0089","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[1,26],\"y_values\":[3,33],\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 107/7, 111/7, 11877/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 107/7, 111/7, 11877/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286089,"parameters":{"x_values":[1,26],"y_values":[3,33],"masses":[[[12,49],[9,49]],[[16,49],[12,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[9,49]],[[16,49],[12,49]]],\"x_values\":[1,26],\"y_values\":[3,33]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0090","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[2,27],\"y_values\":[-3,34],\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 139/7, 90/7, 12510/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 139/7, 90/7, 12510/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286090,"parameters":{"x_values":[2,27],"y_values":[-3,34],"masses":[[[8,49],[6,49]],[[20,49],[15,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[8,49],[6,49]],[[20,49],[15,49]]],\"x_values\":[2,27],\"y_values\":[-3,34]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0091","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[3,28],\"y_values\":[-2,35],\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 46/7, 134/7, 6164/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 46/7, 134/7, 6164/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286091,"parameters":{"x_values":[3,28],"y_values":[-2,35],"masses":[[[18,49],[24,49]],[[3,49],[4,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[18,49],[24,49]],[[3,49],[4,49]]],\"x_values\":[3,28],\"y_values\":[-2,35]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0092","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[4,29],\"y_values\":[-1,36],\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 78/7, 141/7, 10998/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 78/7, 141/7, 10998/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":2,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286092,"parameters":{"x_values":[4,29],"y_values":[-1,36],"masses":[[[15,49],[20,49]],[[6,49],[8,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[15,49],[20,49]],[[6,49],[8,49]]],\"x_values\":[4,29],\"y_values\":[-1,36]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0093","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[5,30],\"y_values\":[0,37],\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 110/7, 148/7, 16280/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 110/7, 148/7, 16280/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286093,"parameters":{"x_values":[5,30],"y_values":[0,37],"masses":[[[12,49],[16,49]],[[9,49],[12,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[16,49]],[[9,49],[12,49]]],\"x_values\":[5,30],\"y_values\":[0,37]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0094","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[6,31],\"y_values\":[1,38],\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 142/7, 155/7, 22010/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 142/7, 155/7, 22010/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286094,"parameters":{"x_values":[6,31],"y_values":[1,38],"masses":[[[9,49],[12,49]],[[12,49],[16,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[9,49],[12,49]],[[12,49],[16,49]]],\"x_values\":[6,31],\"y_values\":[1,38]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0095","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[7,32],\"y_values\":[2,39],\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 174/7, 162/7, 28188/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 174/7, 162/7, 28188/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286095,"parameters":{"x_values":[7,32],"y_values":[2,39],"masses":[[[6,49],[8,49]],[[15,49],[20,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[6,49],[8,49]],[[15,49],[20,49]]],\"x_values\":[7,32],\"y_values\":[2,39]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0096","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[8,33],\"y_values\":[3,40],\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 81/7, 206/7, 16686/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 81/7, 206/7, 16686/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286096,"parameters":{"x_values":[8,33],"y_values":[3,40],"masses":[[[12,49],[30,49]],[[2,49],[5,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[12,49],[30,49]],[[2,49],[5,49]]],\"x_values\":[8,33],\"y_values\":[3,40]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0097","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[9,34],\"y_values\":[-3,41],\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 113/7, 199/7, 22487/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 113/7, 199/7, 22487/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286097,"parameters":{"x_values":[9,34],"y_values":[-3,41],"masses":[[[10,49],[25,49]],[[4,49],[10,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[10,49],[25,49]],[[4,49],[10,49]]],\"x_values\":[9,34],\"y_values\":[-3,41]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0098","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[10,35],\"y_values\":[-2,42],\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 145/7, 206/7, 29870/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 145/7, 206/7, 29870/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286098,"parameters":{"x_values":[10,35],"y_values":[-2,42],"masses":[[[8,49],[20,49]],[[6,49],[15,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[8,49],[20,49]],[[6,49],[15,49]]],\"x_values\":[10,35],\"y_values\":[-2,42]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0099","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[11,36],\"y_values\":[-1,43],\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 177/7, 213/7, 37701/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 177/7, 213/7, 37701/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286099,"parameters":{"x_values":[11,36],"y_values":[-1,43],"masses":[[[6,49],[15,49]],[[8,49],[20,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[6,49],[15,49]],[[8,49],[20,49]]],\"x_values\":[11,36],\"y_values\":[-1,43]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}
{"schema_version":"ocme-question-v0.1","id":"q-20261014-advance-20261028-r06-random-variable-independence-0100","batch_id":"batch-20261014-advance-20261028-r06-random-variable-independence-foundations","language":"zh-Hant","source_mko_ids":["mko-random-variable-independence"],"title_zh":"隨機變數獨立","stem_zh":"全部計算綁定同一完整有限joint law。操作=rvind-zero-covariance；資料={\"x_values\":[12,37],\"y_values\":[0,44],\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]]}。請依 [independent,E_X,E_Y,E_XY,covariance,implication_pass,boundary] 欄序回答。","answer":{"kind":"expression","value":"(是, 209/7, 220/7, 45980/49, 0, 是, zero_covariance_converse_not_claimed)","explanation_zh":"Wave4第二層核對=(是, 209/7, 220/7, 45980/49, 0, 是, zero_covariance_converse_not_claimed)；依完整明示資料重算；完整有限／命名精確切片不自動升格為任意無限定理；全部值對或事件逆像須滿足乘積責任；零協方差不反推獨立且不推出IID或Markov。"},"difficulty":{"band":"undergraduate_core","level":3,"skills":["隨機變數獨立"]},"variant":{"family_id":"qf-mod28-rvind-zero-covariance","seed":202610286100,"parameters":{"x_values":[12,37],"y_values":[0,44],"masses":[[[4,49],[10,49]],[[10,49],[25,49]]],"domain":"Exact_finite_joint_random_variable_independence","operation":"rvind-zero-covariance","semantic_template":"qf-mod28-rvind-zero-covariance/{\"masses\":[[[4,49],[10,49]],[[10,49],[25,49]]],\"x_values\":[12,37],\"y_values\":[0,44]}"}},"verification":{"status":"candidate","checks":["producer_exact_finite_model","producer_source_snapshot_check","producer_substantive_template_deduplication","producer_json_schema"],"warning_zh":"產生端候選須由主端重算；不構成人工審定、任意無限定理、整環／域、Hilbert、無限維算子、mutual independence、有向連通、Lean執行或形式Evidence。"},"provenance":{"generator_role":"question_producer","generator_model":"model_unknown","generated_at":"2026-10-14","contract_version":"ocme-question-factory-v0.1"}}