{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0001","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3efd918a51b7ef9602a73c5fc30dbe4e736cccc37f5d2f26fdd3d40386a9d799","family_id":"qf-random-variable-evaluation","seed":202610055001,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0],\"mapping\":[[0,0]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0}, with event family F=𝒫(S). The total function X:S→R has table {(0, 0)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"0","explanation":"The elementary outcome belongs to S and its unique value is X(0)=0. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0002","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0e0de16e82ff3d26519ea0a1e7ee44d0832ac451bb5eff9dc9cbbd48a84e909b","family_id":"qf-random-variable-evaluation","seed":202610055002,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-1],[1,-1],[2,3],[3,1]],\"operation\":\"rv_evaluate\",\"s\":1}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 3), (3, 1)}. Find X(1), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-1","explanation":"The elementary outcome belongs to S and its unique value is X(1)=-1. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0003","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b588f029f45f5a6757b5bd3b4c6ced6598647643a134c30c9807227dffd1ddd","family_id":"qf-random-variable-evaluation","seed":202610055003,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-2],[1,-2],[2,4],[3,0]],\"operation\":\"rv_evaluate\",\"s\":2}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 4), (3, 0)}. Find X(2), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"4","explanation":"The elementary outcome belongs to S and its unique value is X(2)=4. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0004","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c6b513c5e0cad019eac1c38309f6fd2205349cef2131b89e0c58934d584d80ac","family_id":"qf-random-variable-evaluation","seed":202610055004,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-3],[1,-3],[2,5],[3,5]],\"operation\":\"rv_evaluate\",\"s\":3}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 5), (3, 5)}. Find X(3), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"5","explanation":"The elementary outcome belongs to S and its unique value is X(3)=5. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0005","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f565635e2cf6f275f751712434167d0f41ad6aa7b7b18bf4954eae6ac5261cf3","family_id":"qf-random-variable-evaluation","seed":202610055005,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-4],[1,-4],[2,6],[3,-2]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 6), (3, -2)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-4","explanation":"The elementary outcome belongs to S and its unique value is X(0)=-4. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0006","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a65c411dc0b77662db61f23db66c61385e005f3a66f161f7d5022a791a824de2","family_id":"qf-random-variable-evaluation","seed":202610055006,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-5],[1,-5],[2,7],[3,-3]],\"operation\":\"rv_evaluate\",\"s\":1}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 7), (3, -3)}. Find X(1), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-5","explanation":"The elementary outcome belongs to S and its unique value is X(1)=-5. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0007","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58c893e33b02bc92f95c3956600d1b83bfa57149eba2b2af72a704a940646ba6","family_id":"qf-random-variable-evaluation","seed":202610055007,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-6],[1,-6],[2,8],[3,8]],\"operation\":\"rv_evaluate\",\"s\":2}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 8), (3, 8)}. Find X(2), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"8","explanation":"The elementary outcome belongs to S and its unique value is X(2)=8. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0008","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb0552b50992b2604b7b98b80ded2c8bd0305f97b64c456a9584503c945f4e69","family_id":"qf-random-variable-evaluation","seed":202610055008,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-7],[1,-7],[2,9],[3,-5]],\"operation\":\"rv_evaluate\",\"s\":3}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 9), (3, -5)}. Find X(3), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-5","explanation":"The elementary outcome belongs to S and its unique value is X(3)=-5. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0009","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"04b3a69a0f9bf32c2c02fe20dfec3591d4f92cfbc6bb1ed2a7119426f3815d9a","family_id":"qf-random-variable-evaluation","seed":202610055009,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-8],[1,-8],[2,10],[3,-6]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 10), (3, -6)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-8","explanation":"The elementary outcome belongs to S and its unique value is X(0)=-8. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0010","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0fbf93aa8cc9a37606588d89fc0dfd54b8eb31cd85a511f5dcec29a53e134b8","family_id":"qf-random-variable-evaluation","seed":202610055010,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-9],[1,-9],[2,11],[3,11]],\"operation\":\"rv_evaluate\",\"s\":1}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 11), (3, 11)}. Find X(1), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-9","explanation":"The elementary outcome belongs to S and its unique value is X(1)=-9. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0011","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0b5d45102ce0b61ebf7f9548070f9bb57fdb6ad61d939137ba05d8ec8d557784","family_id":"qf-random-variable-evaluation","seed":202610055011,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-10],[1,-10],[2,12],[3,-8]],\"operation\":\"rv_evaluate\",\"s\":2}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 12), (3, -8)}. Find X(2), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"12","explanation":"The elementary outcome belongs to S and its unique value is X(2)=12. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0012","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c519cc6f1d15ec1bb326d74e231a34f752251a32cc4b6d5950e16a733abd098","family_id":"qf-random-variable-evaluation","seed":202610055012,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-11],[1,-11],[2,13],[3,-9]],\"operation\":\"rv_evaluate\",\"s\":3}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 13), (3, -9)}. Find X(3), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-9","explanation":"The elementary outcome belongs to S and its unique value is X(3)=-9. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0013","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3447cd6bd2b0673ad4e9d85c48532164548d1882fe92b78605aee3719169c9af","family_id":"qf-random-variable-evaluation","seed":202610055013,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-12],[1,-12],[2,14],[3,14]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 14), (3, 14)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-12","explanation":"The elementary outcome belongs to S and its unique value is X(0)=-12. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0014","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0dc82e0166ba5839a4aeee76c12056b6c6b0a7c8b50bf2d39dd2bb6e658bd59a","family_id":"qf-random-variable-evaluation","seed":202610055014,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-13],[1,-13],[2,15],[3,-11]],\"operation\":\"rv_evaluate\",\"s\":1}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 15), (3, -11)}. Find X(1), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-13","explanation":"The elementary outcome belongs to S and its unique value is X(1)=-13. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0015","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90b84624e06cb74133dfec875c31743c40f5c7e8036d310b06e4ce34d9fc6e21","family_id":"qf-random-variable-evaluation","seed":202610055015,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-14],[1,-14],[2,16],[3,-12]],\"operation\":\"rv_evaluate\",\"s\":2}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 16), (3, -12)}. Find X(2), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"16","explanation":"The elementary outcome belongs to S and its unique value is X(2)=16. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0016","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c8529e3771fa1355150c85fcbd491da5620d4d8c9348d8cff76130c35c26e83","family_id":"qf-random-variable-evaluation","seed":202610055016,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-15],[1,-15],[2,17],[3,17]],\"operation\":\"rv_evaluate\",\"s\":3}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 17), (3, 17)}. Find X(3), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"17","explanation":"The elementary outcome belongs to S and its unique value is X(3)=17. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0017","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b5ed5e6a01092a3d31f34a7346696ecaa846db6f65c9843954d9d1c350939f","family_id":"qf-random-variable-evaluation","seed":202610055017,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-16],[1,-16],[2,18],[3,-14]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 18), (3, -14)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-16","explanation":"The elementary outcome belongs to S and its unique value is X(0)=-16. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0018","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ea328069793f317b50eb67506c13fa43820e01bc64d386ff69fc4140c617420","family_id":"qf-random-variable-evaluation","seed":202610055018,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_evaluate\",\"s\":1}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. Find X(1), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-17","explanation":"The elementary outcome belongs to S and its unique value is X(1)=-17. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0019","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b79e84945ed6ecda41c6dc4ca37436bb8dd477803e89b974702deb95095179ef","family_id":"qf-random-variable-evaluation","seed":202610055019,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-18],[1,-18],[2,20],[3,20]],\"operation\":\"rv_evaluate\",\"s\":2}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 20), (3, 20)}. Find X(2), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"20","explanation":"The elementary outcome belongs to S and its unique value is X(2)=20. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0020","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"087a452b58cda07be0be5dc4cd5b274bcec8baeeb432b128a1bb418ef137a9a0","family_id":"qf-random-variable-evaluation","seed":202610055020,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-19],[1,-19],[2,21],[3,-17]],\"operation\":\"rv_evaluate\",\"s\":3}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 21), (3, -17)}. Find X(3), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-17","explanation":"The elementary outcome belongs to S and its unique value is X(3)=-17. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0021","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c63f42209133f34091c97eb77279967e6547cc6374d9dd6ee1c49262d3eeba51","family_id":"qf-random-variable-evaluation","seed":202610055021,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-20],[1,-20],[2,22],[3,-18]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-20","explanation":"The elementary outcome belongs to S and its unique value is X(0)=-20. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0022","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49fbcd306d6d060ce76af28dd23c8f479e15c885930d799b5eef7f3022baf062","family_id":"qf-random-variable-evaluation","seed":202610055022,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-21],[1,-21],[2,23],[3,23]],\"operation\":\"rv_evaluate\",\"s\":1}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 23), (3, 23)}. Find X(1), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-21","explanation":"The elementary outcome belongs to S and its unique value is X(1)=-21. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0023","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ba0a0b346058577be7ada930553e2d7870afb6c0f856eaf247bcb7f4121f871","family_id":"qf-random-variable-evaluation","seed":202610055023,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-22],[1,-22],[2,24],[3,-20]],\"operation\":\"rv_evaluate\",\"s\":2}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 24), (3, -20)}. Find X(2), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"24","explanation":"The elementary outcome belongs to S and its unique value is X(2)=24. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0024","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1216be7e3f079792523b26452a42b53a9b26229859e51d5da3b798be4999008d","family_id":"qf-random-variable-evaluation","seed":202610055024,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-23],[1,-23],[2,25],[3,-21]],\"operation\":\"rv_evaluate\",\"s\":3}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 25), (3, -21)}. Find X(3), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-21","explanation":"The elementary outcome belongs to S and its unique value is X(3)=-21. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0025","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17b95383c8e510160d7b1e7e4947f6aa4843d8417c6069234a693f00055a3edd","family_id":"qf-random-variable-evaluation","seed":202610055025,"semantic_template":"qf-random-variable-evaluation/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-24],[1,-24],[2,26],[3,26]],\"operation\":\"rv_evaluate\",\"s\":0}"},"fields":{"title":"Random-Variable Value at an Elementary Outcome","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 26), (3, 26)}. Find X(0), reading only the function value without assigning unspecified probabilities.","answer":{"kind":"short_text","value":"-24","explanation":"The elementary outcome belongs to S and its unique value is X(0)=-24. No unspecified probability law or equiprobability is inferred."},"skills":["Random-Variable Value at an Elementary Outcome"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0026","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191935077f813daeda78239f4c3cdea1abc1a9b92e011bf4265c07b8a8799792","family_id":"qf-random-variable-threshold-event","seed":202610055026,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0],\"mapping\":[[0,0]],\"operation\":\"rv_threshold\",\"t\":-1}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0}, with event family F=𝒫(S). The total function X:S→R has table {(0, 0)}. List the event {s∈S:X(s)≥-1} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0}","explanation":"Check the inclusive threshold at each outcome to obtain {0}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0027","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fd75133154eb0dcd1de7a61e3fc81bd7725aba21496921cb07ccf18948d18c9","family_id":"qf-random-variable-threshold-event","seed":202610055027,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-1],[1,-1],[2,3],[3,1]],\"operation\":\"rv_threshold\",\"t\":4}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 3), (3, 1)}. List the event {s∈S:X(s)≥4} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"∅","explanation":"Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0028","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3a1f4c6fbd24b90be2fbd37e76c24822e6592ce5bb04cc566f15e8f2dab0286","family_id":"qf-random-variable-threshold-event","seed":202610055028,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-2],[1,-2],[2,4],[3,0]],\"operation\":\"rv_threshold\",\"t\":0}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 4), (3, 0)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0029","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd69015ce64effc19e19461b3aee1118d0753cf040ad2b69049e6656ad51c736","family_id":"qf-random-variable-threshold-event","seed":202610055029,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-3],[1,-3],[2,5],[3,5]],\"operation\":\"rv_threshold\",\"t\":-3}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 5), (3, 5)}. List the event {s∈S:X(s)≥-3} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0030","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"def3530fc6bba6f8bbb5a402d89e5017478b2652579e78f4f20fb805ad7748d9","family_id":"qf-random-variable-threshold-event","seed":202610055030,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-4],[1,-4],[2,6],[3,-2]],\"operation\":\"rv_threshold\",\"t\":-5}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 6), (3, -2)}. List the event {s∈S:X(s)≥-5} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0031","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"64a2f5c25cc6aebd9db6747d850c3393c401a7d97204cb1689929fb0cbdde2da","family_id":"qf-random-variable-threshold-event","seed":202610055031,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-5],[1,-5],[2,7],[3,-3]],\"operation\":\"rv_threshold\",\"t\":8}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 7), (3, -3)}. List the event {s∈S:X(s)≥8} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"∅","explanation":"Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0032","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d48da83d2215e2e36f35e65a259821bc8f64bc7e1c21c5bf46eb3a839edcb966","family_id":"qf-random-variable-threshold-event","seed":202610055032,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-6],[1,-6],[2,8],[3,8]],\"operation\":\"rv_threshold\",\"t\":0}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 8), (3, 8)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0033","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d9ec9540db22fd71bbc5cd8c317ff0847090d09acf0b45a7bf80a8b08d31d793","family_id":"qf-random-variable-threshold-event","seed":202610055033,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-7],[1,-7],[2,9],[3,-5]],\"operation\":\"rv_threshold\",\"t\":-7}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 9), (3, -5)}. List the event {s∈S:X(s)≥-7} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0034","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c4520e2f18f8bbadc1eeeabf67090b9ab8febe3d2a59047125db3a0f8eda0ee","family_id":"qf-random-variable-threshold-event","seed":202610055034,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-8],[1,-8],[2,10],[3,-6]],\"operation\":\"rv_threshold\",\"t\":-9}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 10), (3, -6)}. List the event {s∈S:X(s)≥-9} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0035","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa92eef300448f0d52fb88e6f3d66382211104645da8d76bc53f38288402cc05","family_id":"qf-random-variable-threshold-event","seed":202610055035,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-9],[1,-9],[2,11],[3,11]],\"operation\":\"rv_threshold\",\"t\":12}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 11), (3, 11)}. List the event {s∈S:X(s)≥12} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"∅","explanation":"Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0036","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5571e87ae7a0d7de46bbcb83fc79f6d922ee63bb3bb0eebbad7e59c1f0910200","family_id":"qf-random-variable-threshold-event","seed":202610055036,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-10],[1,-10],[2,12],[3,-8]],\"operation\":\"rv_threshold\",\"t\":0}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 12), (3, -8)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{2}","explanation":"Check the inclusive threshold at each outcome to obtain {2}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0037","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e3c7a6865decf3f8bc42fb32500682d97f9ab473e8fb93158d1a06ab7c49cb76","family_id":"qf-random-variable-threshold-event","seed":202610055037,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-11],[1,-11],[2,13],[3,-9]],\"operation\":\"rv_threshold\",\"t\":-11}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 13), (3, -9)}. List the event {s∈S:X(s)≥-11} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0038","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa42c4cc678084907c679fde6373e61dcacaf1de7ce158607d3195e58af5c662","family_id":"qf-random-variable-threshold-event","seed":202610055038,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-12],[1,-12],[2,14],[3,14]],\"operation\":\"rv_threshold\",\"t\":-13}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 14), (3, 14)}. List the event {s∈S:X(s)≥-13} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0039","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1558aca63f324e08433fbc192456fdb816c94dac99ddf0b078bf414b1099ef79","family_id":"qf-random-variable-threshold-event","seed":202610055039,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-13],[1,-13],[2,15],[3,-11]],\"operation\":\"rv_threshold\",\"t\":16}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 15), (3, -11)}. List the event {s∈S:X(s)≥16} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"∅","explanation":"Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0040","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad1b42ed0027ced7a7ca69b4dde41efa9c4a077c41a5336ab2c579c7edb97cb0","family_id":"qf-random-variable-threshold-event","seed":202610055040,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-14],[1,-14],[2,16],[3,-12]],\"operation\":\"rv_threshold\",\"t\":0}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 16), (3, -12)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{2}","explanation":"Check the inclusive threshold at each outcome to obtain {2}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0041","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"30e042584ad6e54ab005c2e279bd4530e87b727ff45c6ba47737e0477fd2cc72","family_id":"qf-random-variable-threshold-event","seed":202610055041,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-15],[1,-15],[2,17],[3,17]],\"operation\":\"rv_threshold\",\"t\":-15}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 17), (3, 17)}. List the event {s∈S:X(s)≥-15} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0042","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e01966a5b5db9355ff3051f91ac2f30d3afd3d57f424f8b846e65fdc3940c04","family_id":"qf-random-variable-threshold-event","seed":202610055042,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-16],[1,-16],[2,18],[3,-14]],\"operation\":\"rv_threshold\",\"t\":-17}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 18), (3, -14)}. List the event {s∈S:X(s)≥-17} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0043","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4114d2931c899a08737373557cd7b3cafcaacd95677b51aef5b6133d6db0582a","family_id":"qf-random-variable-threshold-event","seed":202610055043,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_threshold\",\"t\":20}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. List the event {s∈S:X(s)≥20} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"∅","explanation":"Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0044","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"81271cdade752174bc04a7b1362bc95d1726989984c0b3639ed50a2c79dbfd83","family_id":"qf-random-variable-threshold-event","seed":202610055044,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-18],[1,-18],[2,20],[3,20]],\"operation\":\"rv_threshold\",\"t\":0}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 20), (3, 20)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0045","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"964ff6e407c31d8bb40794ab7b247d91f743425f60f742e07da35ef08e3207dc","family_id":"qf-random-variable-threshold-event","seed":202610055045,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-19],[1,-19],[2,21],[3,-17]],\"operation\":\"rv_threshold\",\"t\":-19}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 21), (3, -17)}. List the event {s∈S:X(s)≥-19} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0046","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16e387cc56df6645cfae9d4e0a60234342ecd7743c92f5d9a98d838ef551df5a","family_id":"qf-random-variable-threshold-event","seed":202610055046,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-20],[1,-20],[2,22],[3,-18]],\"operation\":\"rv_threshold\",\"t\":-21}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. List the event {s∈S:X(s)≥-21} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0047","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49dd524a18f39412023279aac262d466954f8977ed6593b304903fda479c5502","family_id":"qf-random-variable-threshold-event","seed":202610055047,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-21],[1,-21],[2,23],[3,23]],\"operation\":\"rv_threshold\",\"t\":24}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 23), (3, 23)}. List the event {s∈S:X(s)≥24} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"∅","explanation":"Check the inclusive threshold at each outcome to obtain ∅. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0048","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65c0a954b7cf32300d0ace691624c863711ed59ff05b0f0ced31c88316338025","family_id":"qf-random-variable-threshold-event","seed":202610055048,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-22],[1,-22],[2,24],[3,-20]],\"operation\":\"rv_threshold\",\"t\":0}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 24), (3, -20)}. List the event {s∈S:X(s)≥0} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{2}","explanation":"Check the inclusive threshold at each outcome to obtain {2}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0049","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d391594595e4b623af39eec40ec96d1f71a7c88ddb708b6e818ccc35fb75efc7","family_id":"qf-random-variable-threshold-event","seed":202610055049,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-23],[1,-23],[2,25],[3,-21]],\"operation\":\"rv_threshold\",\"t\":-23}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 25), (3, -21)}. List the event {s∈S:X(s)≥-23} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0050","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f3cd4b7c5846065b799aa3536920405a09a57ddb9cd8c0bbb5683e9a43add11","family_id":"qf-random-variable-threshold-event","seed":202610055050,"semantic_template":"qf-random-variable-threshold-event/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-24],[1,-24],[2,26],[3,26]],\"operation\":\"rv_threshold\",\"t\":-25}"},"fields":{"title":"Preimage Event of a Threshold Condition","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 26), (3, 26)}. List the event {s∈S:X(s)≥-25} in increasing order. Return elementary outcomes, not a numerical probability.","answer":{"kind":"short_text","value":"{0, 1, 2, 3}","explanation":"Check the inclusive threshold at each outcome to obtain {0, 1, 2, 3}. F is the full powerset, so this subset is an event; without weights, no event probability is asserted."},"skills":["Preimage Event of a Threshold Condition"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0051","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7703beef36d25168717cf132d56a1ea37c663373b8038230a1d18eb3e11b322","family_id":"qf-random-variable-range","seed":202610055051,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0],\"mapping\":[[0,0]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0}, with event family F=𝒫(S). The total function X:S→R has table {(0, 0)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{0}","explanation":"The range is {0}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0052","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c0ffadc4e8f13420b8c3053e6598619b9dc4eb6045f274dd64499d9da245a49","family_id":"qf-random-variable-range","seed":202610055052,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-1],[1,-1],[2,3],[3,1]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 3), (3, 1)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-1, 1, 3}","explanation":"The range is {-1, 1, 3}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0053","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e722e6b6379d9305fb68c742c2cfe2ecd10460e667077dd849e766bdca71108","family_id":"qf-random-variable-range","seed":202610055053,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-2],[1,-2],[2,4],[3,0]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 4), (3, 0)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-2, 0, 4}","explanation":"The range is {-2, 0, 4}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0054","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03e9070ad541f21879ab7206e9da8aa4c394275a1331ceef4b319b2bae745504","family_id":"qf-random-variable-range","seed":202610055054,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-3],[1,-3],[2,5],[3,5]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 5), (3, 5)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-3, 5}","explanation":"The range is {-3, 5}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0055","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb083ec4b9c36044db8052aa547a77f499bff6fa9a265a76e9908d8877196e36","family_id":"qf-random-variable-range","seed":202610055055,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-4],[1,-4],[2,6],[3,-2]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 6), (3, -2)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-4, -2, 6}","explanation":"The range is {-4, -2, 6}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0056","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90741ab893f41923087e0ad3cd36a080bd3f05f1527aa1712c8616d5f695e8ae","family_id":"qf-random-variable-range","seed":202610055056,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-5],[1,-5],[2,7],[3,-3]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 7), (3, -3)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-5, -3, 7}","explanation":"The range is {-5, -3, 7}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0057","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a9b07a95949f0fec54062e4e2120f2ccf99452054264f00f7f830494099aba1","family_id":"qf-random-variable-range","seed":202610055057,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-6],[1,-6],[2,8],[3,8]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 8), (3, 8)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-6, 8}","explanation":"The range is {-6, 8}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0058","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2834cd5fa29e7cbd5362ae058ff6ec0a192b92e78a3618e1f1ebe7b97d34c96","family_id":"qf-random-variable-range","seed":202610055058,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-7],[1,-7],[2,9],[3,-5]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 9), (3, -5)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-7, -5, 9}","explanation":"The range is {-7, -5, 9}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0059","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c65c8538e4665819720dc0def911d007d8c9272d06d7bd334f8ee3fd61ed495","family_id":"qf-random-variable-range","seed":202610055059,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-8],[1,-8],[2,10],[3,-6]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 10), (3, -6)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-8, -6, 10}","explanation":"The range is {-8, -6, 10}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0060","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"220e02cef24ebc16b5b41e43e7eb8bcffbb18db850dbb53c2ce289241f5648c8","family_id":"qf-random-variable-range","seed":202610055060,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-9],[1,-9],[2,11],[3,11]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 11), (3, 11)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-9, 11}","explanation":"The range is {-9, 11}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0061","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9cff5aa59b15cf8eea182f729de9a52a0fad083d11e407a753af2870d460c340","family_id":"qf-random-variable-range","seed":202610055061,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-10],[1,-10],[2,12],[3,-8]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 12), (3, -8)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-10, -8, 12}","explanation":"The range is {-10, -8, 12}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0062","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41ca2298761923e2fc598a6a3a37ac1bcc4217293cec499aa15bb6dc241ad833","family_id":"qf-random-variable-range","seed":202610055062,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-11],[1,-11],[2,13],[3,-9]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 13), (3, -9)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-11, -9, 13}","explanation":"The range is {-11, -9, 13}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0063","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65a043323f7857c85982d07f3d0353a9f814629d42090aec698018cb1c28d239","family_id":"qf-random-variable-range","seed":202610055063,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-12],[1,-12],[2,14],[3,14]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 14), (3, 14)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-12, 14}","explanation":"The range is {-12, 14}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0064","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be35032eb913486961ecadea31cce87048e231c86e24366380c2bb016deba2ed","family_id":"qf-random-variable-range","seed":202610055064,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-13],[1,-13],[2,15],[3,-11]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 15), (3, -11)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-13, -11, 15}","explanation":"The range is {-13, -11, 15}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0065","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4a2592e8aa1dc5c28a6c67711c631bf791ba939abec34be88ef8054bfceabd3","family_id":"qf-random-variable-range","seed":202610055065,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-14],[1,-14],[2,16],[3,-12]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 16), (3, -12)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-14, -12, 16}","explanation":"The range is {-14, -12, 16}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0066","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76634de300dd1ceade0711522d1e211edb5c6664260e4c752bf1beb0dd925d89","family_id":"qf-random-variable-range","seed":202610055066,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-15],[1,-15],[2,17],[3,17]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 17), (3, 17)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-15, 17}","explanation":"The range is {-15, 17}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0067","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9f7575cf5c10f44f96dc960f9abb661cef3134c9e1ff65d643e9317ef3c72a1","family_id":"qf-random-variable-range","seed":202610055067,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-16],[1,-16],[2,18],[3,-14]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 18), (3, -14)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-16, -14, 18}","explanation":"The range is {-16, -14, 18}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0068","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"191b4b0707540c9e85745faeebefb704f0ce24dddbc386241739ab3274bb10ff","family_id":"qf-random-variable-range","seed":202610055068,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-17],[1,-17],[2,19],[3,-15]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 19), (3, -15)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-17, -15, 19}","explanation":"The range is {-17, -15, 19}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0069","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"947b7391eb5b34ee2afaa69ff3aa37840110ee18f5496ae00b4f95bb773d678a","family_id":"qf-random-variable-range","seed":202610055069,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-18],[1,-18],[2,20],[3,20]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 20), (3, 20)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-18, 20}","explanation":"The range is {-18, 20}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0070","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14264bcc97ce21afff2fe80e53830c461597d98e9f5b82d2143a3deff1528b09","family_id":"qf-random-variable-range","seed":202610055070,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-19],[1,-19],[2,21],[3,-17]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 21), (3, -17)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-19, -17, 21}","explanation":"The range is {-19, -17, 21}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0071","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ce4a55de4319b515631c59628c4b44be7203a7f7c8042be9a84f67c67d4b8cd","family_id":"qf-random-variable-range","seed":202610055071,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-20],[1,-20],[2,22],[3,-18]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 22), (3, -18)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-20, -18, 22}","explanation":"The range is {-20, -18, 22}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0072","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62ab0a1d5b0b9a3ca9b0c55e1611e4e5154bb1572ce876c758d91974fb7b35b7","family_id":"qf-random-variable-range","seed":202610055072,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-21],[1,-21],[2,23],[3,23]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 23), (3, 23)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-21, 23}","explanation":"The range is {-21, 23}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0073","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"065db576d0e359aad7946f81f731327ed244fa7839842780d97835e647bcb040","family_id":"qf-random-variable-range","seed":202610055073,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-22],[1,-22],[2,24],[3,-20]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 24), (3, -20)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-22, -20, 24}","explanation":"The range is {-22, -20, 24}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0074","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27de720c02b66ef8cda0b23f4eedd6451f795207bf48a78b5bdc12409b4e4015","family_id":"qf-random-variable-range","seed":202610055074,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-23],[1,-23],[2,25],[3,-21]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 25), (3, -21)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-23, -21, 25}","explanation":"The range is {-23, -21, 25}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0075","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f668d1f34979882de1986ecb62c67f06d78e92dca4f26dd8ee9cbbd745a14223","family_id":"qf-random-variable-range","seed":202610055075,"semantic_template":"qf-random-variable-range/{\"F\":\"powerset(S)\",\"S\":[0,1,2,3],\"mapping\":[[0,-24],[1,-24],[2,26],[3,26]],\"operation\":\"rv_range\"}"},"fields":{"title":"Attained Range of a Random Variable","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 26), (3, 26)}. List the attained range X[S] in increasing order with each output value once.","answer":{"kind":"short_text","value":"{-24, 26}","explanation":"The range is {-24, 26}, a set of attained output values rather than elementary outcomes or probability weights. Removing duplicate output values does not remove their preimage weights from a probability calculation."},"skills":["Attained Range of a Random Variable"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0076","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"204e5cf33168e0b33dec2d7807e28e79e30779577aa6d2b2ddec6502807520d6","family_id":"qf-random-variable-point-mass","seed":202610055076,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-1\",\"1/2\"],[\"2\",\"1/2\"]],\"operation\":\"rv_point_mass\",\"t\":-1}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -1), (1, -1), (2, 2), (3, 999)}. In the displayed order of S, the nonnegative probability weights are [1/(6), 2/(6), 3/(6), 0/(6)], summing exactly to one. Find P(X=-1) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"1/2","explanation":"Add the weights of every outcome whose value equals -1, giving P(X=-1)=1/2. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0077","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f2bc03a3bb0fac10344a01ae252b61049efae1d68b0f22b9144abf2d70169a57","family_id":"qf-random-variable-point-mass","seed":202610055077,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-2\",\"4/7\"],[\"3\",\"3/7\"]],\"operation\":\"rv_point_mass\",\"t\":3}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -2), (1, -2), (2, 3), (3, 1000)}. In the displayed order of S, the nonnegative probability weights are [1/(7), 3/(7), 3/(7), 0/(7)], summing exactly to one. Find P(X=3) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"3/7","explanation":"Add the weights of every outcome whose value equals 3, giving P(X=3)=3/7. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0078","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ebed713d5f66f0d88953b288902c0e51083505d8b7fa759176bacdc39e00129b","family_id":"qf-random-variable-point-mass","seed":202610055078,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-3\",\"5/8\"],[\"4\",\"3/8\"]],\"operation\":\"rv_point_mass\",\"t\":1001}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -3), (1, -3), (2, 4), (3, 1001)}. In the displayed order of S, the nonnegative probability weights are [1/(8), 4/(8), 3/(8), 0/(8)], summing exactly to one. Find P(X=1001) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 1001, giving P(X=1001)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0079","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"96e249d0c83d76f21e0b8d09da683f7bb73c15f20e6ece8454b94e64fb498e5c","family_id":"qf-random-variable-point-mass","seed":202610055079,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-4\",\"2/3\"],[\"5\",\"1/3\"]],\"operation\":\"rv_point_mass\",\"t\":0}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -4), (1, -4), (2, 5), (3, 1002)}. In the displayed order of S, the nonnegative probability weights are [1/(9), 5/(9), 3/(9), 0/(9)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0080","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f322663da300a6b4da62bc3769323cddaf1ee8baa5157ec0831b25a6cd95b81","family_id":"qf-random-variable-point-mass","seed":202610055080,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-5\",\"7/10\"],[\"6\",\"3/10\"]],\"operation\":\"rv_point_mass\",\"t\":-5}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -5), (1, -5), (2, 6), (3, 1003)}. In the displayed order of S, the nonnegative probability weights are [1/(10), 6/(10), 3/(10), 0/(10)], summing exactly to one. Find P(X=-5) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"7/10","explanation":"Add the weights of every outcome whose value equals -5, giving P(X=-5)=7/10. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0081","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3da45d645c9b756c31e200348a8eb27cf41b587bab1530506ee37a3c04769d95","family_id":"qf-random-variable-point-mass","seed":202610055081,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-6\",\"8/11\"],[\"7\",\"3/11\"]],\"operation\":\"rv_point_mass\",\"t\":7}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -6), (1, -6), (2, 7), (3, 1004)}. In the displayed order of S, the nonnegative probability weights are [1/(11), 7/(11), 3/(11), 0/(11)], summing exactly to one. Find P(X=7) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"3/11","explanation":"Add the weights of every outcome whose value equals 7, giving P(X=7)=3/11. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0082","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a0e1acd8169b72505f6880355f4116c0129eb166c7d709b7f68c0ab089f6680","family_id":"qf-random-variable-point-mass","seed":202610055082,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-7\",\"3/4\"],[\"8\",\"1/4\"]],\"operation\":\"rv_point_mass\",\"t\":1005}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -7), (1, -7), (2, 8), (3, 1005)}. In the displayed order of S, the nonnegative probability weights are [1/(12), 8/(12), 3/(12), 0/(12)], summing exactly to one. Find P(X=1005) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 1005, giving P(X=1005)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0083","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17e5fee62c69394e036fa6476870dec98deecf9ecfc5321b1c6050bd681e5385","family_id":"qf-random-variable-point-mass","seed":202610055083,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-8\",\"10/13\"],[\"9\",\"3/13\"]],\"operation\":\"rv_point_mass\",\"t\":0}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -8), (1, -8), (2, 9), (3, 1006)}. In the displayed order of S, the nonnegative probability weights are [1/(13), 9/(13), 3/(13), 0/(13)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0084","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91e4e059bbca0596380ce4de61d7e1301c073f9567fd1388bf674be00bd1ff8f","family_id":"qf-random-variable-point-mass","seed":202610055084,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-9\",\"11/14\"],[\"10\",\"3/14\"]],\"operation\":\"rv_point_mass\",\"t\":-9}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -9), (1, -9), (2, 10), (3, 1007)}. In the displayed order of S, the nonnegative probability weights are [1/(14), 10/(14), 3/(14), 0/(14)], summing exactly to one. Find P(X=-9) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"11/14","explanation":"Add the weights of every outcome whose value equals -9, giving P(X=-9)=11/14. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0085","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ad9e7541dc5b56110803b714d3c32271034eabd114db3180d4fda197eaf0117","family_id":"qf-random-variable-point-mass","seed":202610055085,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-10\",\"4/5\"],[\"11\",\"1/5\"]],\"operation\":\"rv_point_mass\",\"t\":11}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -10), (1, -10), (2, 11), (3, 1008)}. In the displayed order of S, the nonnegative probability weights are [1/(15), 11/(15), 3/(15), 0/(15)], summing exactly to one. Find P(X=11) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"1/5","explanation":"Add the weights of every outcome whose value equals 11, giving P(X=11)=1/5. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0086","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5f58fdee0b3428e300ec25f8b652878189524f0292101dd6b48356d30326d600","family_id":"qf-random-variable-point-mass","seed":202610055086,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-11\",\"13/16\"],[\"12\",\"3/16\"]],\"operation\":\"rv_point_mass\",\"t\":1009}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -11), (1, -11), (2, 12), (3, 1009)}. In the displayed order of S, the nonnegative probability weights are [1/(16), 12/(16), 3/(16), 0/(16)], summing exactly to one. Find P(X=1009) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 1009, giving P(X=1009)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0087","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61501efeb3cc83afd5c94f2c628a439a1e09c6057e97294e64343967c2f58099","family_id":"qf-random-variable-point-mass","seed":202610055087,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-12\",\"14/17\"],[\"13\",\"3/17\"]],\"operation\":\"rv_point_mass\",\"t\":0}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -12), (1, -12), (2, 13), (3, 1010)}. In the displayed order of S, the nonnegative probability weights are [1/(17), 13/(17), 3/(17), 0/(17)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0088","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"287b80eba13f477801ee3c2e24f37451b63250d6fb99c749fdc362eb8fd6db56","family_id":"qf-random-variable-point-mass","seed":202610055088,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-13\",\"5/6\"],[\"14\",\"1/6\"]],\"operation\":\"rv_point_mass\",\"t\":-13}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -13), (1, -13), (2, 14), (3, 1011)}. In the displayed order of S, the nonnegative probability weights are [1/(18), 14/(18), 3/(18), 0/(18)], summing exactly to one. Find P(X=-13) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"5/6","explanation":"Add the weights of every outcome whose value equals -13, giving P(X=-13)=5/6. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0089","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0804835e6a2aaed08e80550bba5f348e061503b9419ea933e7c34337a7b73e5f","family_id":"qf-random-variable-point-mass","seed":202610055089,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-14\",\"16/19\"],[\"15\",\"3/19\"]],\"operation\":\"rv_point_mass\",\"t\":15}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -14), (1, -14), (2, 15), (3, 1012)}. In the displayed order of S, the nonnegative probability weights are [1/(19), 15/(19), 3/(19), 0/(19)], summing exactly to one. Find P(X=15) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"3/19","explanation":"Add the weights of every outcome whose value equals 15, giving P(X=15)=3/19. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0090","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe77bd4fecc0a59b099a1779c471356f58a3e7baa9ce825cf70338a624e79231","family_id":"qf-random-variable-point-mass","seed":202610055090,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-15\",\"17/20\"],[\"16\",\"3/20\"]],\"operation\":\"rv_point_mass\",\"t\":1013}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -15), (1, -15), (2, 16), (3, 1013)}. In the displayed order of S, the nonnegative probability weights are [1/(20), 16/(20), 3/(20), 0/(20)], summing exactly to one. Find P(X=1013) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 1013, giving P(X=1013)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0091","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b24811971192902fcb0929e6eacbf0915cd6fab17f1690f1bc4ca82be3bced56","family_id":"qf-random-variable-point-mass","seed":202610055091,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-16\",\"6/7\"],[\"17\",\"1/7\"]],\"operation\":\"rv_point_mass\",\"t\":0}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -16), (1, -16), (2, 17), (3, 1014)}. In the displayed order of S, the nonnegative probability weights are [1/(21), 17/(21), 3/(21), 0/(21)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0092","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"684473fcf00fff9d93e10dea5c422dac4d28d903b1d0e398d6feb6206703c4b8","family_id":"qf-random-variable-point-mass","seed":202610055092,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-17\",\"19/22\"],[\"18\",\"3/22\"]],\"operation\":\"rv_point_mass\",\"t\":-17}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -17), (1, -17), (2, 18), (3, 1015)}. In the displayed order of S, the nonnegative probability weights are [1/(22), 18/(22), 3/(22), 0/(22)], summing exactly to one. Find P(X=-17) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"19/22","explanation":"Add the weights of every outcome whose value equals -17, giving P(X=-17)=19/22. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0093","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeebf161672c293fdcb793d7cb53683cdb3433e3d75ca620475079fe6c921de9","family_id":"qf-random-variable-point-mass","seed":202610055093,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-18\",\"20/23\"],[\"19\",\"3/23\"]],\"operation\":\"rv_point_mass\",\"t\":19}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -18), (1, -18), (2, 19), (3, 1016)}. In the displayed order of S, the nonnegative probability weights are [1/(23), 19/(23), 3/(23), 0/(23)], summing exactly to one. Find P(X=19) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"3/23","explanation":"Add the weights of every outcome whose value equals 19, giving P(X=19)=3/23. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0094","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d4645b3d27b30b622e0d095065c7ab09681e0bfa3576ecb01677f1eb591a0662","family_id":"qf-random-variable-point-mass","seed":202610055094,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-19\",\"7/8\"],[\"20\",\"1/8\"]],\"operation\":\"rv_point_mass\",\"t\":1017}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -19), (1, -19), (2, 20), (3, 1017)}. In the displayed order of S, the nonnegative probability weights are [1/(24), 20/(24), 3/(24), 0/(24)], summing exactly to one. Find P(X=1017) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 1017, giving P(X=1017)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0095","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"869b4dba3f1fe262e3ab2a2265c936602040e5bf383dd54b13c8d2bb0cb5faa5","family_id":"qf-random-variable-point-mass","seed":202610055095,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-20\",\"22/25\"],[\"21\",\"3/25\"]],\"operation\":\"rv_point_mass\",\"t\":0}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -20), (1, -20), (2, 21), (3, 1018)}. In the displayed order of S, the nonnegative probability weights are [1/(25), 21/(25), 3/(25), 0/(25)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0096","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c674aee911f0a740a6c77930c838aa3d238b0985a0e3ebded80c41f93f7ec7cb","family_id":"qf-random-variable-point-mass","seed":202610055096,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-21\",\"23/26\"],[\"22\",\"3/26\"]],\"operation\":\"rv_point_mass\",\"t\":-21}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -21), (1, -21), (2, 22), (3, 1019)}. In the displayed order of S, the nonnegative probability weights are [1/(26), 22/(26), 3/(26), 0/(26)], summing exactly to one. Find P(X=-21) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"23/26","explanation":"Add the weights of every outcome whose value equals -21, giving P(X=-21)=23/26. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0097","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f93f42afee0e38f93128a82a3ae5620facc3e9e72a3b0f4b7bc8ea7c2502ae4c","family_id":"qf-random-variable-point-mass","seed":202610055097,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-22\",\"8/9\"],[\"23\",\"1/9\"]],\"operation\":\"rv_point_mass\",\"t\":23}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -22), (1, -22), (2, 23), (3, 1020)}. In the displayed order of S, the nonnegative probability weights are [1/(27), 23/(27), 3/(27), 0/(27)], summing exactly to one. Find P(X=23) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"1/9","explanation":"Add the weights of every outcome whose value equals 23, giving P(X=23)=1/9. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0098","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3b8ab58264c67979b7cb19212188ab4a8418bd7ba7327d1498e714e0946a4276","family_id":"qf-random-variable-point-mass","seed":202610055098,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-23\",\"25/28\"],[\"24\",\"3/28\"]],\"operation\":\"rv_point_mass\",\"t\":1021}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -23), (1, -23), (2, 24), (3, 1021)}. In the displayed order of S, the nonnegative probability weights are [1/(28), 24/(28), 3/(28), 0/(28)], summing exactly to one. Find P(X=1021) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 1021, giving P(X=1021)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0099","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3e619a010d2f4899c2fa9b1c5795af4b918c8b74e9c0a70fca10f5a3e01ebe03","family_id":"qf-random-variable-point-mass","seed":202610055099,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-24\",\"26/29\"],[\"25\",\"3/29\"]],\"operation\":\"rv_point_mass\",\"t\":0}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -24), (1, -24), (2, 25), (3, 1022)}. In the displayed order of S, the nonnegative probability weights are [1/(29), 25/(29), 3/(29), 0/(29)], summing exactly to one. Find P(X=0) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"0/1","explanation":"Add the weights of every outcome whose value equals 0, giving P(X=0)=0/1. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261004-advance-20261005-r05-random-variable-0100","batch_id":"batch-20261004-advance-20261005-r05-random-variable-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0908bf244af02228d45aed8d50f54f030efd2ec8cfbf2954488683d9c4608c9e","family_id":"qf-random-variable-point-mass","seed":202610055100,"semantic_template":"qf-random-variable-point-mass/{\"F\":\"powerset(S)\",\"law\":[[\"-25\",\"9/10\"],[\"26\",\"1/10\"]],\"operation\":\"rv_point_mass\",\"t\":-25}"},"fields":{"title":"Finite Point Probability of a Repeated Value","stem":"The nonempty finite sample space is S={0, 1, 2, 3}, with event family F=𝒫(S). The total function X:S→R has table {(0, -25), (1, -25), (2, 26), (3, 1023)}. In the displayed order of S, the nonnegative probability weights are [1/(30), 26/(30), 3/(30), 0/(30)], summing exactly to one. Find P(X=-25) exactly as a reduced fraction p/q with positive denominator.","answer":{"kind":"expression","value":"9/10","explanation":"Add the weights of every outcome whose value equals -25, giving P(X=-25)=9/10. Zero-weight preimages add no mass; an unattained value has probability 0/1. Range cardinality is not a denominator."},"skills":["Finite Point Probability of a Repeated Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}