{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r05-expectation-0001","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07ca3a3e84c27d48f420771f576b3184bd4f1f9eadd282c86f62ed7941c05194","family_id":"qf-density10-expectation-contributions","seed":202610105001,"semantic_template":"qf-density10-expectation-contributions/[[-3,\"1/3\"],[\"zero\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 99, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-1/1, 0/1, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-1/1, 0/1, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0002","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"43711f5ca17803705883d84f61a3cdff7dcc42e062a09b75a4589eb6e4c70317","family_id":"qf-density10-expectation-contributions","seed":202610105002,"semantic_template":"qf-density10-expectation-contributions/[[\"zero\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[0/1, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [0/1, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0003","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f4be771c36c91ae858c0486a560f41e02171b4e14a63029ca29325eac844bf2","family_id":"qf-density10-expectation-contributions","seed":202610105003,"semantic_template":"qf-density10-expectation-contributions/[[-2,\"1/2\"],[2,\"1/3\"],[1,\"1/6\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[3, 2, 1], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-1/1, 2/3, 1/6]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-1/1, 2/3, 1/6]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0004","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d149b2ca0185703bfd9b0ed1051a7f103cdfadc31371e36a7a05bd853e88c75","family_id":"qf-density10-expectation-contributions","seed":202610105004,"semantic_template":"qf-density10-expectation-contributions/[[-3,\"4/7\"],[2,\"2/7\"],[-1,\"1/7\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-12/7, 4/7, -1/7]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-12/7, 4/7, -1/7]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0005","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df04a351d193d5f018bd40dc3a6186c25eeaeb18a506ca6957fc6f2e3c466349","family_id":"qf-density10-expectation-contributions","seed":202610105005,"semantic_template":"qf-density10-expectation-contributions/[[-4,\"5/8\"],[2,\"1/4\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-5/2, 1/2, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-5/2, 1/2, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0006","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b318f4fe7b882b8786b385dd8297174dad44c2b9e21ee4a3c35542d9601e9683","family_id":"qf-density10-expectation-contributions","seed":202610105006,"semantic_template":"qf-density10-expectation-contributions/[[-5,\"2/3\"],[2,\"2/9\"],[1,\"1/9\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-10/3, 4/9, 1/9]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-10/3, 4/9, 1/9]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0007","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8fb688c70d8e7276c91489d845f1b6112b8ce7e26f96ad57717999ef169d2984","family_id":"qf-density10-expectation-contributions","seed":202610105007,"semantic_template":"qf-density10-expectation-contributions/[[-6,\"7/10\"],[2,\"1/5\"],[-1,\"1/10\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-6, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-21/5, 2/5, -1/10]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-21/5, 2/5, -1/10]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0008","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f88368f5fd8adb82ff2af7b3c401b05ec57196410989f0714ba9d4f509c1fa64","family_id":"qf-density10-expectation-contributions","seed":202610105008,"semantic_template":"qf-density10-expectation-contributions/[[-7,\"8/11\"],[2,\"2/11\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-56/11, 4/11, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-56/11, 4/11, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0009","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83867579f5b256ec06c91d905bdf8c1eba9cceb49d1850dd40257487f6c36a4","family_id":"qf-density10-expectation-contributions","seed":202610105009,"semantic_template":"qf-density10-expectation-contributions/[[-8,\"3/4\"],[2,\"1/6\"],[1,\"1/12\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-6/1, 1/3, 1/12]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-6/1, 1/3, 1/12]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0010","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"65c43e18fc01742c4e0ed55e4bddf816faa8dd179e699c3291dc1f03e16ba74c","family_id":"qf-density10-expectation-contributions","seed":202610105010,"semantic_template":"qf-density10-expectation-contributions/[[-9,\"10/13\"],[2,\"2/13\"],[-1,\"1/13\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-9, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-90/13, 4/13, -1/13]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-90/13, 4/13, -1/13]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0011","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21069773d298b5ca699961bb63678caac043edffa75f17323fbcb5ab7c7e9cc2","family_id":"qf-density10-expectation-contributions","seed":202610105011,"semantic_template":"qf-density10-expectation-contributions/[[-10,\"11/14\"],[2,\"1/7\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-55/7, 2/7, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-55/7, 2/7, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0012","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dcda0b11de1781aa712eb8276b73b4274196afe3908679b89f2b21be27217b6b","family_id":"qf-density10-expectation-contributions","seed":202610105012,"semantic_template":"qf-density10-expectation-contributions/[[-11,\"4/5\"],[2,\"2/15\"],[1,\"1/15\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-44/5, 4/15, 1/15]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-44/5, 4/15, 1/15]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0013","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee961b5123c02ba0e97c8bdace2843139e8b1f3a7a749c6a256c1c83a364944f","family_id":"qf-density10-expectation-contributions","seed":202610105013,"semantic_template":"qf-density10-expectation-contributions/[[-12,\"13/16\"],[2,\"1/8\"],[-1,\"1/16\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-39/4, 1/4, -1/16]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-39/4, 1/4, -1/16]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0014","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a9850b91e535391cb6edfcf1fd7d30c888872aea18f0cc53598eb1dc420d76f","family_id":"qf-density10-expectation-contributions","seed":202610105014,"semantic_template":"qf-density10-expectation-contributions/[[-13,\"14/17\"],[2,\"2/17\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-182/17, 4/17, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-182/17, 4/17, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0015","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2ebc9c5bbad2c7aff9fd9ce37c702d99db030b69372b7c1beecb62f7c5d2b25e","family_id":"qf-density10-expectation-contributions","seed":202610105015,"semantic_template":"qf-density10-expectation-contributions/[[-14,\"5/6\"],[2,\"1/9\"],[1,\"1/18\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-35/3, 2/9, 1/18]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-35/3, 2/9, 1/18]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0016","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7507da0e049a7f7a3584a578a9a89c8a55b7cdf37606fd63204a4886318620e0","family_id":"qf-density10-expectation-contributions","seed":202610105016,"semantic_template":"qf-density10-expectation-contributions/[[-15,\"16/19\"],[2,\"2/19\"],[-1,\"1/19\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-15, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-240/19, 4/19, -1/19]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-240/19, 4/19, -1/19]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0017","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"799b31ae36f1e56107e3f767cc86b7e035fb5d69acd2eded538a86df6b62cd66","family_id":"qf-density10-expectation-contributions","seed":202610105017,"semantic_template":"qf-density10-expectation-contributions/[[-16,\"17/20\"],[2,\"1/10\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-68/5, 1/5, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-68/5, 1/5, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0018","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0fde941591a66081618753d86de19bbc4692b0d3d11a5de3afe1ba670439539d","family_id":"qf-density10-expectation-contributions","seed":202610105018,"semantic_template":"qf-density10-expectation-contributions/[[-17,\"6/7\"],[2,\"2/21\"],[1,\"1/21\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-102/7, 4/21, 1/21]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-102/7, 4/21, 1/21]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0019","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"667e5149eff46d0a06ae5f13b9b714b307d4671fd4bb69bc075be53e14e304ad","family_id":"qf-density10-expectation-contributions","seed":202610105019,"semantic_template":"qf-density10-expectation-contributions/[[-18,\"19/22\"],[2,\"1/11\"],[-1,\"1/22\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-18, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-171/11, 2/11, -1/22]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-171/11, 2/11, -1/22]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0020","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01f111b6dce4f854a75bce1608c666b3bcee2b6b867820d1415c1d7c2a01099b","family_id":"qf-density10-expectation-contributions","seed":202610105020,"semantic_template":"qf-density10-expectation-contributions/[[-19,\"20/23\"],[2,\"2/23\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-380/23, 4/23, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-380/23, 4/23, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0021","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"243774f2e05ed0d201a917c035eb59aa76b6db98471969712d746c27e2a587e9","family_id":"qf-density10-expectation-contributions","seed":202610105021,"semantic_template":"qf-density10-expectation-contributions/[[-20,\"7/8\"],[2,\"1/12\"],[1,\"1/24\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-35/2, 1/6, 1/24]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-35/2, 1/6, 1/24]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0022","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e00844422c0f66dd60844a93e4b520f12aa9c59bbd93d7a1eeb85358d680fb43","family_id":"qf-density10-expectation-contributions","seed":202610105022,"semantic_template":"qf-density10-expectation-contributions/[[-21,\"22/25\"],[2,\"2/25\"],[-1,\"1/25\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-462/25, 4/25, -1/25]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-462/25, 4/25, -1/25]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0023","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1566ea5cc102a032b8c96cff1702bdde98b4eca7fa9d821e111e471c24ef56ef","family_id":"qf-density10-expectation-contributions","seed":202610105023,"semantic_template":"qf-density10-expectation-contributions/[[-22,\"23/26\"],[2,\"1/13\"],[\"zero\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-253/13, 2/13, 0/1]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-253/13, 2/13, 0/1]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0024","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fd1497d47307918c5eb273281097f4b73e60ccdea6d18d18fc3d3afe3391f5f","family_id":"qf-density10-expectation-contributions","seed":202610105024,"semantic_template":"qf-density10-expectation-contributions/[[-23,\"8/9\"],[2,\"2/27\"],[1,\"1/27\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-184/9, 4/27, 1/27]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-184/9, 4/27, 1/27]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0025","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c033700d5927d0b39936350014d4e0b6f827854717adc0a037ee42e16c14ac96","family_id":"qf-density10-expectation-contributions","seed":202610105025,"semantic_template":"qf-density10-expectation-contributions/[[-24,\"25/28\"],[2,\"1/14\"],[-1,\"1/28\"]]"},"fields":{"title":"Every Weighted Expectation Contribution","stem":"S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[-150/7, 1/7, -1/28]","explanation":"Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-150/7, 1/7, -1/28]."},"skills":["Every Weighted Expectation Contribution"],"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-20261008-advance-20261010-r05-expectation-0026","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f480a9f107e1ffd7c7507515a92038a6709d334c171ba16e78e75c0225baf2a9","family_id":"qf-density10-expectation-difference","seed":202610105026,"semantic_template":"qf-density10-expectation-difference/[[\"0\",\"1/1\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[4, 99, 4] in index order and Y=[4, -50, 4] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is 0/1."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0027","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bea7ec7ee2602059dbe7d5802e868c32ef74faf014905033e9e82ee0fc4e59e0","family_id":"qf-density10-expectation-difference","seed":202610105027,"semantic_template":"qf-density10-expectation-difference/[[\"-2\",\"1/2\"],[\"2\",\"1/2\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 2] in index order and Y=[2, 0] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is 0/1."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0028","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eb1e8019957dc48f34dca4a9e4a71d21539bf8a3a016ccf47dbffa3f22ca861c","family_id":"qf-density10-expectation-difference","seed":202610105028,"semantic_template":"qf-density10-expectation-difference/[[\"-3\",\"1/2\"],[\"3\",\"1/3\"],[\"5\",\"1/6\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[3, 2, 1], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 2, 3] in index order and Y=[1, -1, -2] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"1/3","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is 1/3."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0029","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"182240890fc15718a64f1400e5b3859135cbc6f445ef2568c56853dd05cdb27d","family_id":"qf-density10-expectation-difference","seed":202610105029,"semantic_template":"qf-density10-expectation-difference/[[\"-4\",\"4/7\"],[\"3\",\"2/7\"],[\"7\",\"1/7\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 2, 4] in index order and Y=[1, -1, -3] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-3/7","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -3/7."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0030","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd4c74049728732662ad96bc65e181bd268ca0bbd52ce6694d26554572da916f","family_id":"qf-density10-expectation-difference","seed":202610105030,"semantic_template":"qf-density10-expectation-difference/[[\"-5\",\"5/8\"],[\"3\",\"1/4\"],[\"9\",\"1/8\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 5] in index order and Y=[1, -1, -4] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-5/4","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -5/4."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0031","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11c59aa4933b64087dea75cf1995229dee4500eb27e786ca541081cdb07a4ddb","family_id":"qf-density10-expectation-difference","seed":202610105031,"semantic_template":"qf-density10-expectation-difference/[[\"-6\",\"2/3\"],[\"11\",\"1/9\"],[\"3\",\"2/9\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 6] in index order and Y=[1, -1, -5] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-19/9","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -19/9."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0032","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef1e4a216398db4db1a90f84d605918a18b675bdb31ae32a3c31623581c76a7","family_id":"qf-density10-expectation-difference","seed":202610105032,"semantic_template":"qf-density10-expectation-difference/[[\"-7\",\"7/10\"],[\"13\",\"1/10\"],[\"3\",\"1/5\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-6, 2, 7] in index order and Y=[1, -1, -6] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-3/1","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -3/1."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0033","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1504d5c771a5088910ed3ff2d5ab4533ca88c714054de0ca8a4d36b11fe8f197","family_id":"qf-density10-expectation-difference","seed":202610105033,"semantic_template":"qf-density10-expectation-difference/[[\"-8\",\"8/11\"],[\"15\",\"1/11\"],[\"3\",\"2/11\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 8] in index order and Y=[1, -1, -7] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-43/11","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -43/11."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0034","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"475f40630c240f699a9a9f86575adc85c8f9eda9e0f8a7ca0b67e72ec4476503","family_id":"qf-density10-expectation-difference","seed":202610105034,"semantic_template":"qf-density10-expectation-difference/[[\"-9\",\"3/4\"],[\"17\",\"1/12\"],[\"3\",\"1/6\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 9] in index order and Y=[1, -1, -8] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-29/6","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -29/6."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0035","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"27701b73bb63fc1ae2f9a2a5fe5252e161fa2865bb40368bb4f198e17d12d1e8","family_id":"qf-density10-expectation-difference","seed":202610105035,"semantic_template":"qf-density10-expectation-difference/[[\"-10\",\"10/13\"],[\"19\",\"1/13\"],[\"3\",\"2/13\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-9, 2, 10] in index order and Y=[1, -1, -9] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-75/13","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -75/13."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0036","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"48eab5e71e07efe2c5838b3d8d63781f68acc1d3d4f2721816319bd254623dae","family_id":"qf-density10-expectation-difference","seed":202610105036,"semantic_template":"qf-density10-expectation-difference/[[\"-11\",\"11/14\"],[\"21\",\"1/14\"],[\"3\",\"1/7\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 11] in index order and Y=[1, -1, -10] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-47/7","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -47/7."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0037","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11219f973efde8755f056f522d67e6d30dae670926c07e7e39f5be73236a1d5f","family_id":"qf-density10-expectation-difference","seed":202610105037,"semantic_template":"qf-density10-expectation-difference/[[\"-12\",\"4/5\"],[\"23\",\"1/15\"],[\"3\",\"2/15\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 12] in index order and Y=[1, -1, -11] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-23/3","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -23/3."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0038","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"817689e9fb674b4747136416d45b0ea65112d828346576fb6737402a1195732a","family_id":"qf-density10-expectation-difference","seed":202610105038,"semantic_template":"qf-density10-expectation-difference/[[\"-13\",\"13/16\"],[\"25\",\"1/16\"],[\"3\",\"1/8\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[1, -1, -12] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-69/8","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -69/8."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0039","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2954bcba9d946e5fdfa5515c762e6c2692a2f777da21d1cb2768492d7b5319c5","family_id":"qf-density10-expectation-difference","seed":202610105039,"semantic_template":"qf-density10-expectation-difference/[[\"-14\",\"14/17\"],[\"27\",\"1/17\"],[\"3\",\"2/17\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, 2, 14] in index order and Y=[1, -1, -13] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-163/17","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -163/17."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0040","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9edab3eebbe245abc127f043ac0e94370ee48c8e26e8d9014218145b1fde476d","family_id":"qf-density10-expectation-difference","seed":202610105040,"semantic_template":"qf-density10-expectation-difference/[[\"-15\",\"5/6\"],[\"29\",\"1/18\"],[\"3\",\"1/9\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 15] in index order and Y=[1, -1, -14] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-95/9","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -95/9."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0041","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfefa20b6d44d4963dbe8041c0a31a5a86518c5015aa0c1b1c520cd543f43265","family_id":"qf-density10-expectation-difference","seed":202610105041,"semantic_template":"qf-density10-expectation-difference/[[\"-16\",\"16/19\"],[\"3\",\"2/19\"],[\"31\",\"1/19\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-15, 2, 16] in index order and Y=[1, -1, -15] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-219/19","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -219/19."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0042","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fcf0bb5b12152b0ba5beb8379109360c6dc3749740a1c048346a2838647df2c8","family_id":"qf-density10-expectation-difference","seed":202610105042,"semantic_template":"qf-density10-expectation-difference/[[\"-17\",\"17/20\"],[\"3\",\"1/10\"],[\"33\",\"1/20\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, 2, 17] in index order and Y=[1, -1, -16] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-25/2","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -25/2."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0043","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"690bb862f1fd3d6e985f514db41423069de14e932a0363eab7e0e8271c5d4a48","family_id":"qf-density10-expectation-difference","seed":202610105043,"semantic_template":"qf-density10-expectation-difference/[[\"-18\",\"6/7\"],[\"3\",\"2/21\"],[\"35\",\"1/21\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 2, 18] in index order and Y=[1, -1, -17] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-283/21","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -283/21."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0044","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"864db38fbd4fcf2e3ddaf87c5185d1a2c8daab2ce6fa7f3c4d629e8e1b90159d","family_id":"qf-density10-expectation-difference","seed":202610105044,"semantic_template":"qf-density10-expectation-difference/[[\"-19\",\"19/22\"],[\"3\",\"1/11\"],[\"37\",\"1/22\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-18, 2, 19] in index order and Y=[1, -1, -18] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-159/11","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -159/11."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0045","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4a05aa9d3c8472d5291defb5e55fe3109b6c346a4f517de77f493420fe156d8","family_id":"qf-density10-expectation-difference","seed":202610105045,"semantic_template":"qf-density10-expectation-difference/[[\"-20\",\"20/23\"],[\"3\",\"2/23\"],[\"39\",\"1/23\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 20] in index order and Y=[1, -1, -19] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-355/23","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -355/23."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0046","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"942f7a3f2c057161adea5a7cbf8446b01bdadc366c88a125f628c8bebb9ab375","family_id":"qf-density10-expectation-difference","seed":202610105046,"semantic_template":"qf-density10-expectation-difference/[[\"-21\",\"7/8\"],[\"3\",\"1/12\"],[\"41\",\"1/24\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[1, -1, -20] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-197/12","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -197/12."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0047","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e68c4cd128b48c6bc696a8293b0d46d85d27c11446df2469e32dc3d61f9ae1b","family_id":"qf-density10-expectation-difference","seed":202610105047,"semantic_template":"qf-density10-expectation-difference/[[\"-22\",\"22/25\"],[\"3\",\"2/25\"],[\"43\",\"1/25\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, 22] in index order and Y=[1, -1, -21] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-87/5","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -87/5."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0048","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"34baaf45932dbe5af5653e1dfdace8ea3b350399e1540d2ad42d457fba80d37e","family_id":"qf-density10-expectation-difference","seed":202610105048,"semantic_template":"qf-density10-expectation-difference/[[\"-23\",\"23/26\"],[\"3\",\"1/13\"],[\"45\",\"1/26\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 23] in index order and Y=[1, -1, -22] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-239/13","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -239/13."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0049","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"03dab7460d95c9720815630079823f3f9ba814ca476eb1644e1246efac709485","family_id":"qf-density10-expectation-difference","seed":202610105049,"semantic_template":"qf-density10-expectation-difference/[[\"-24\",\"8/9\"],[\"3\",\"2/27\"],[\"47\",\"1/27\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] in index order and Y=[1, -1, -23] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-523/27","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -523/27."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0050","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8c6bc317ad6cb18d7f32a30da65008f9c8f4ddc2463ca6ee8de4f3681cf88fd5","family_id":"qf-density10-expectation-difference","seed":202610105050,"semantic_template":"qf-density10-expectation-difference/[[\"-25\",\"25/28\"],[\"3\",\"1/14\"],[\"49\",\"1/28\"]]"},"fields":{"title":"Expectation Difference on a Shared Model","stem":"S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, 25] in index order and Y=[1, -1, -24] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"-285/14","explanation":"Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -285/14."},"skills":["Expectation Difference on a Shared Model"],"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-20261008-advance-20261010-r05-expectation-0051","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1775f4f2f983698e75f9b48a782efec922875e85941fcae4a723daba38be9a6f","family_id":"qf-density10-expectation-product-pair","seed":202610105051,"semantic_template":"qf-density10-expectation-product-pair/[[0,0,\"1/2\"],[1,1,\"1/2\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 1] in index order and Y=[0, 1] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/2, 1/4)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (1/2, 1/4)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0052","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c17b8a91de1411dd6d0b5fc7fecef58b0bc1183a57d6918e48ae159f3b41b08","family_id":"qf-density10-expectation-product-pair","seed":202610105052,"semantic_template":"qf-density10-expectation-product-pair/[[-1,3,\"1/3\"],[2,3,\"2/3\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,1}, F=P(S), masses=[1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[3, 3] in index order and Y=[-1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3/1, 3/1)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (3/1, 3/1)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0053","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"039aef51f33324fbeef6bd03acfc15845ea1b6bd37f576e455dc43cf43051960","family_id":"qf-density10-expectation-product-pair","seed":202610105053,"semantic_template":"qf-density10-expectation-product-pair/[[-2,1,\"1/3\"],[1,-2,\"2/3\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 99, 1] in index order and Y=[1, -100, -2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2/1, 0/1)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-2/1, 0/1)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0054","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cad68d32e9fb6e97d5419da25cf15c2d34965054eaf19e188a7d9a188596e0a0","family_id":"qf-density10-expectation-product-pair","seed":202610105054,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/7\"],[2,4,\"1/7\"],[5,-3,\"4/7\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 2, 4] in index order and Y=[5, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-8/1, -80/49)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-8/1, -80/49)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0055","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e2e6d5642868f8256404ca4602648a2e0be48412e6385dbde47921f948b4005c","family_id":"qf-density10-expectation-product-pair","seed":202610105055,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/4\"],[2,5,\"1/8\"],[6,-4,\"5/8\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 5] in index order and Y=[6, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-57/4, -165/32)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-57/4, -165/32)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0056","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35035f04ba3baae6091b5bec1dc3bad348d54838d3f7eefc7e7b48ad4f822328","family_id":"qf-density10-expectation-product-pair","seed":202610105056,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/9\"],[2,6,\"1/9\"],[7,-5,\"2/3\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 6] in index order and Y=[7, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-202/9, -280/27)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-202/9, -280/27)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0057","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a23c0bb22e6325ed3b1cc98e976b45ecb9332fe4cc4df3b19fb65c981e23091","family_id":"qf-density10-expectation-product-pair","seed":202610105057,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/5\"],[2,7,\"1/10\"],[8,-6,\"7/10\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-6, 2, 7] in index order and Y=[8, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-163/5, -434/25)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-163/5, -434/25)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0058","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b616ead50d818bf4ef09482f2afc08da56debd88437ad83e9674fd2e9f127ce9","family_id":"qf-density10-expectation-product-pair","seed":202610105058,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/11\"],[2,8,\"1/11\"],[9,-7,\"8/11\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 8] in index order and Y=[9, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-492/11, -288/11)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-492/11, -288/11)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0059","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d0a26fa35d9501ad7d1cca45b339ac62b39c5215c9863a5a54c32ccbe50ff839","family_id":"qf-density10-expectation-product-pair","seed":202610105059,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/6\"],[10,-8,\"3/4\"],[2,9,\"1/12\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 9] in index order and Y=[10, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-353/6, -295/8)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-353/6, -295/8)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0060","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0d21b21abb153452a6c4e65ed472913b5e339bab08fde8d212eb591ac3af1476","family_id":"qf-density10-expectation-product-pair","seed":202610105060,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/13\"],[11,-9,\"10/13\"],[2,10,\"1/13\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-9, 2, 10] in index order and Y=[11, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-974/13, -8360/169)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-974/13, -8360/169)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0061","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1c1a58edc4c0379cdc5c8dff6b9dd166a74904ca99b58b5c64212a6039cefc9","family_id":"qf-density10-expectation-product-pair","seed":202610105061,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/7\"],[12,-10,\"11/14\"],[2,11,\"1/14\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 11] in index order and Y=[12, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-93/1, -3135/49)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-93/1, -3135/49)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0062","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"092ccaf4e0bb26c6c8a5b9e92373ccffea9242117becc6999719ca1f9a055700","family_id":"qf-density10-expectation-product-pair","seed":202610105062,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/15\"],[13,-11,\"4/5\"],[2,12,\"1/15\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 12] in index order and Y=[13, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1696/15, -6032/75)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1696/15, -6032/75)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0063","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c0d9398c2f1cc378550223b116ba7aea55720928b751dfe0c43a6a76b7b29d1","family_id":"qf-density10-expectation-product-pair","seed":202610105063,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/8\"],[14,-12,\"13/16\"],[2,13,\"1/16\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[14, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1081/8, -12649/128)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1081/8, -12649/128)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0064","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b149575cd9eabc62b2bcd9a9d528eba33b4412a55ec9f5a9ed66358ce8bd1c8","family_id":"qf-density10-expectation-product-pair","seed":202610105064,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/17\"],[15,-13,\"14/17\"],[2,14,\"1/17\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, 2, 14] in index order and Y=[15, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2706/17, -34440/289)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-2706/17, -34440/289)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0065","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c9d657cb83eb2211f231ff764a5708285249e7dccdbf9ed452436f291223f21","family_id":"qf-density10-expectation-product-pair","seed":202610105065,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/9\"],[16,-14,\"5/6\"],[2,15,\"1/18\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 15] in index order and Y=[16, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1667/9, -3820/27)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1667/9, -3820/27)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0066","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42da4e7be7f0b28b1ac6d3e952b2bb5047a7b4ec981fda17a13adb794baf0242","family_id":"qf-density10-expectation-product-pair","seed":202610105066,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/19\"],[17,-15,\"16/19\"],[2,16,\"1/19\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-15, 2, 16] in index order and Y=[17, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4052/19, -59840/361)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-4052/19, -59840/361)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0067","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2d027c647320a2f08884853728f13c9922e2ddcc360cc09bc4050a2c7fdfc2dd","family_id":"qf-density10-expectation-product-pair","seed":202610105067,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/10\"],[18,-16,\"17/20\"],[2,17,\"1/20\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, 2, 17] in index order and Y=[18, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-2433/10, -38403/200)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-2433/10, -38403/200)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0068","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c44562d0170ecf26487e57c20d53ac5f791680d7647a53d116a1b33493504c60","family_id":"qf-density10-expectation-product-pair","seed":202610105068,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/21\"],[19,-17,\"6/7\"],[2,18,\"1/21\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 2, 18] in index order and Y=[19, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-826/3, -10792/49)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-826/3, -10792/49)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0069","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"abb0f4508a7654b64cc0da22cb8c5f7e1be654bce1251dfbb90bf7f876be1494","family_id":"qf-density10-expectation-product-pair","seed":202610105069,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/11\"],[2,19,\"1/22\"],[20,-18,\"19/22\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-18, 2, 19] in index order and Y=[20, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-3403/11, -2755/11)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-3403/11, -2755/11)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0070","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5a8fec505979554078a6086f79fb22f2135daf8aefd9b1a463452d75abedb13e","family_id":"qf-density10-expectation-product-pair","seed":202610105070,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/23\"],[2,20,\"1/23\"],[21,-19,\"20/23\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 20] in index order and Y=[21, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-7944/23, -149520/529)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-7944/23, -149520/529)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0071","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e944a106fcc6ed6133ccabe108bc8e5c2e5476fa0875003f9698041147427ed","family_id":"qf-density10-expectation-product-pair","seed":202610105071,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/12\"],[2,21,\"1/24\"],[22,-20,\"7/8\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[22, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-4601/12, -30415/96)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-4601/12, -30415/96)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0072","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0ff6daa03eb98c1a7ed69ef3f6f67024d4b669f6dd9f281c504c0fe6d689b34","family_id":"qf-density10-expectation-product-pair","seed":202610105072,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/25\"],[2,22,\"1/25\"],[23,-21,\"22/25\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, 22] in index order and Y=[23, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-10586/25, -220616/625)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-10586/25, -220616/625)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0073","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed26b447d9c9b1ecd8cc94b07767c3202f9e190af48097a741a34b5dc434c089","family_id":"qf-density10-expectation-product-pair","seed":202610105073,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/13\"],[2,23,\"1/26\"],[24,-22,\"23/26\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 23] in index order and Y=[24, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-6051/13, -66102/169)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-6051/13, -66102/169)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0074","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf96db79d668667d0734da616d8fed570fff9a5b5d5a9848bf25544cc182577e","family_id":"qf-density10-expectation-product-pair","seed":202610105074,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"2/27\"],[2,24,\"1/27\"],[25,-23,\"8/9\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] in index order and Y=[25, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-13756/27, -104800/243)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-13756/27, -104800/243)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0075","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8f877f200836f81ef74a2ea818199ccba933dc0817aa043e4b27f883d11cd43","family_id":"qf-density10-expectation-product-pair","seed":202610105075,"semantic_template":"qf-density10-expectation-product-pair/[[-1,2,\"1/14\"],[2,25,\"1/28\"],[26,-24,\"25/28\"]]"},"fields":{"title":"Product Expectation versus Product of Means","stem":"S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, 25] in index order and Y=[26, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(-1111/2, -185575/392)","explanation":"Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1111/2, -185575/392)."},"skills":["Product Expectation versus Product of Means"],"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-20261008-advance-20261010-r05-expectation-0076","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"876cac95cf6316f173a91419dc2cc9deab95c2541e2939b88f5c6b4d0b3e28ae","family_id":"qf-density10-expectation-missing-value","seed":202610105076,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[\"zero\"],[3,\"2/3\"]],\"1/3\",\"1/1\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 99, 3] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=1/1. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-3/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -3/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0077","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6af40853c660df938335ad5569c92d06d5a5618a0f71d132118be43f6ff40b7","family_id":"qf-density10-expectation-missing-value","seed":202610105077,"semantic_template":"qf-density10-expectation-missing-value/[0,[null],\"1/1\",\"0/1\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,0}, F=P(S), masses=[1], D=1, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=0/1. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"0/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 0/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0078","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fb026a422c5e4836aeb6492c660b977e0eb1649bba44d6c6c48db6accbb9102","family_id":"qf-density10-expectation-missing-value","seed":202610105078,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-2,\"1/2\"],[4,\"1/3\"],null],\"1/6\",\"1/3\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[3, 2, 1], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 4, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=1/3. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"0/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 0/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0079","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"900767b2262b770878361dd80e2b7611274a7079d7b872b174d802bdc187e195","family_id":"qf-density10-expectation-missing-value","seed":202610105079,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[5,\"2/7\"],[-1,\"1/7\"]],\"4/7\",\"-3/7\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 5, -1] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-3/7. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-3/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -3/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0080","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9f1df1a5d5b30c7cf079d2dee565219f11559e25cf6634c4e00bf07383f21c4b","family_id":"qf-density10-expectation-missing-value","seed":202610105080,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-4,\"5/8\"],null,[-2,\"1/8\"]],\"1/4\",\"-5/4\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, null, -2] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-5/4. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"6/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 6/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0081","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"905568f83a1393de2ae1a55e5bc717c3300b193bf7bce4459d35f2af22f3d8e7","family_id":"qf-density10-expectation-missing-value","seed":202610105081,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-5,\"2/3\"],[7,\"2/9\"],null],\"1/9\",\"-19/9\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 7, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-19/9. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-3/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -3/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0082","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3c5c6074d46c63cd5396582abb65c13cc928d06e9e4536a3c3799064bd88bb3c","family_id":"qf-density10-expectation-missing-value","seed":202610105082,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[8,\"1/5\"],[-4,\"1/10\"]],\"7/10\",\"-3/1\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 8, -4] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-3/1. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-6/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -6/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0083","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c4b88b72cd182afaa22f923fe9a97e7349f8eb293400e908613fcae3c95839c8","family_id":"qf-density10-expectation-missing-value","seed":202610105083,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-7,\"8/11\"],null,[-5,\"1/11\"]],\"2/11\",\"-43/11\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, null, -5] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-43/11. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"9/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 9/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0084","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"83553a98afcafe108943a308389514e0339494024a686dcec0a4e3ef090df164","family_id":"qf-density10-expectation-missing-value","seed":202610105084,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-8,\"3/4\"],[10,\"1/6\"],null],\"1/12\",\"-29/6\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 10, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-29/6. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-6/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -6/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0085","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1a29a9624460e85634b81efddd4adaa0744e0d312a70667b8fad59b7badfc3b","family_id":"qf-density10-expectation-missing-value","seed":202610105085,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[11,\"2/13\"],[-7,\"1/13\"]],\"10/13\",\"-75/13\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 11, -7] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-75/13. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-9/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -9/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0086","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eeff760ebe816f61b8843fd5055d5922a4d742d9e95a3295e0aa96460e72d9a0","family_id":"qf-density10-expectation-missing-value","seed":202610105086,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-10,\"11/14\"],null,[-8,\"1/14\"]],\"1/7\",\"-47/7\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, null, -8] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-47/7. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"12/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 12/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0087","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a21771f1de7c9f5d10790d13d53b2c0a663e1f0d4a5e8255f75803c65744ef5b","family_id":"qf-density10-expectation-missing-value","seed":202610105087,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-11,\"4/5\"],[13,\"2/15\"],null],\"1/15\",\"-23/3\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 13, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-23/3. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-9/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -9/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0088","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f83ece90961282b3c60ec4beda7094e7a932cfa97bafc41054abf8e76d08566c","family_id":"qf-density10-expectation-missing-value","seed":202610105088,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[14,\"1/8\"],[-10,\"1/16\"]],\"13/16\",\"-69/8\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 14, -10] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-69/8. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-12/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -12/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0089","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c138aa49c70c903cdfbb89de5d50414009a5c208a9ff2c0ae7db92859b04a290","family_id":"qf-density10-expectation-missing-value","seed":202610105089,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-13,\"14/17\"],null,[-11,\"1/17\"]],\"2/17\",\"-163/17\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, null, -11] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-163/17. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"15/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 15/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0090","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f38d8217aca5a149db886c4fabad0b068b0980c0dd79581193a5b5a2c82d4909","family_id":"qf-density10-expectation-missing-value","seed":202610105090,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-14,\"5/6\"],[16,\"1/9\"],null],\"1/18\",\"-95/9\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 16, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-95/9. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-12/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -12/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0091","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f4975cecfda3e989c0508ab6f00bc3804b4df470d162acba725867e21644fbb","family_id":"qf-density10-expectation-missing-value","seed":202610105091,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[17,\"2/19\"],[-13,\"1/19\"]],\"16/19\",\"-219/19\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 17, -13] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-219/19. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-15/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -15/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0092","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a909f12d8428a053686d5bfdbf993bdc2ca449a55362ff9f92f6156c921610d","family_id":"qf-density10-expectation-missing-value","seed":202610105092,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-16,\"17/20\"],null,[-14,\"1/20\"]],\"1/10\",\"-25/2\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, null, -14] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-25/2. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"18/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 18/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0093","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8022e229e07fd434f758a717be99d8927bfa377d9851433403afb6c7d90aaa8","family_id":"qf-density10-expectation-missing-value","seed":202610105093,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-17,\"6/7\"],[19,\"2/21\"],null],\"1/21\",\"-283/21\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 19, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-283/21. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-15/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -15/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0094","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9efa8202f94a9eaa9d6612bdb45e945b1898a1e12460375bff442563a865a12e","family_id":"qf-density10-expectation-missing-value","seed":202610105094,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[20,\"1/11\"],[-16,\"1/22\"]],\"19/22\",\"-159/11\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 20, -16] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-159/11. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-18/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -18/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0095","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49c1be998aaff9ab3e417cb0747cec67cedaf0fd5e8bf3a4fa9a197ab6740800","family_id":"qf-density10-expectation-missing-value","seed":202610105095,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-19,\"20/23\"],null,[-17,\"1/23\"]],\"2/23\",\"-355/23\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, null, -17] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-355/23. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"21/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 21/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0096","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf5f4cad659c2b2729a830429e7dfce627a7f01d138f787be497649350fb74a0","family_id":"qf-density10-expectation-missing-value","seed":202610105096,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-20,\"7/8\"],[22,\"1/12\"],null],\"1/24\",\"-197/12\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 22, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-197/12. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-18/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -18/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0097","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6d2a93f93b66cb3de6df63c9ef1f2f1d511a2118e33589ff07521684f19d8e9d","family_id":"qf-density10-expectation-missing-value","seed":202610105097,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[23,\"2/25\"],[-19,\"1/25\"]],\"22/25\",\"-87/5\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 23, -19] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-87/5. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-21/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -21/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0098","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70e954b0e5b9a651ebc10e5cff83b93a75b56ff61ac0c165b79680a8ba500443","family_id":"qf-density10-expectation-missing-value","seed":202610105098,"semantic_template":"qf-density10-expectation-missing-value/[1,[[-22,\"23/26\"],null,[-20,\"1/26\"]],\"1/13\",\"-239/13\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, null, -20] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-239/13. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"24/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 24/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0099","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a784fc59215befee2110a121414c2a2e59e8a8baa4170e89b367ac7464a91d14","family_id":"qf-density10-expectation-missing-value","seed":202610105099,"semantic_template":"qf-density10-expectation-missing-value/[2,[[-23,\"8/9\"],[25,\"2/27\"],null],\"1/27\",\"-523/27\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 25, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-523/27. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-21/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -21/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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-20261008-advance-20261010-r05-expectation-0100","batch_id":"batch-20261008-advance-20261010-r05-expectation-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01988e690fe2ab3f1b92a3b320feb92fe3227820f2b7c79b8967f22b9e777726","family_id":"qf-density10-expectation-missing-value","seed":202610105100,"semantic_template":"qf-density10-expectation-missing-value/[0,[null,[26,\"1/14\"],[-22,\"1/28\"]],\"25/28\",\"-285/14\"]"},"fields":{"title":"Positive-Mass Value Recovered from a Mean","stem":"S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 26, -22] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-285/14. Find that slot’s integer value, written as p/1.","answer":{"kind":"expression","value":"-24/1","explanation":"Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -24/1."},"skills":["Positive-Mass Value Recovered from a Mean"],"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."}}