{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261008-advance-20261010-r06-variance-0001","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bae4ca96196edc042102ab314a9f541571a30d6c301fbbddcdcddd53f9bb91a7","family_id":"qf-density10-variance-contributions","seed":202610106001,"semantic_template":"qf-density10-variance-contributions/[\"zero_variance_contributions\",3]"},"fields":{"title":"Every Centered Variance 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=[4, 99, 4] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[0/1, 0/1, 0/1]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [0/1, 0/1, 0/1]."},"skills":["Every Centered Variance 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-r06-variance-0002","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"278ed790b739aad551185ef9206b6c2b41c67f9599d8ed1660f1cb9a8ff89e4b","family_id":"qf-density10-variance-contributions","seed":202610106002,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"1/2\"],[\"-2\",\"1/2\"]]"},"fields":{"title":"Every Centered Variance 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, 2] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[1/2, 1/2]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1/2, 1/2]."},"skills":["Every Centered Variance 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-r06-variance-0003","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b483c0bbd6d0edb4d467f5f0fa1d927ee474365a51e48a4094005f02aa9ce34","family_id":"qf-density10-variance-contributions","seed":202610106003,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"1/2\"],[\"-4\",\"1/3\"],[\"-5\",\"1/6\"]]"},"fields":{"title":"Every Centered Variance 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, 3] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[169/72, 121/108, 289/216]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [169/72, 121/108, 289/216]."},"skills":["Every Centered Variance 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-r06-variance-0004","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"288b2255a48b7decb0699f94e7a4e08641526b3656dcbb07919f9444a60a17a8","family_id":"qf-density10-variance-contributions","seed":202610106004,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"4/7\"],[\"-5\",\"2/7\"],[\"-7\",\"1/7\"]]"},"fields":{"title":"Every Centered Variance 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, 4] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[1156/343, 648/343, 1024/343]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1156/343, 648/343, 1024/343]."},"skills":["Every Centered Variance 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-r06-variance-0005","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"db557f3ced045444029bfe2823a1b4d31128fbfc8a7b35e856fd10c408fb1aba","family_id":"qf-density10-variance-contributions","seed":202610106005,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"5/8\"],[\"-6\",\"1/4\"],[\"-9\",\"1/8\"]]"},"fields":{"title":"Every Centered Variance 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, 5] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[2205/512, 729/256, 2601/512]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [2205/512, 729/256, 2601/512]."},"skills":["Every Centered Variance 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-r06-variance-0006","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dafdc32e15c6d647eead1e8e69fce8c509125ab35d6af0d6634bb4ab12354eb","family_id":"qf-density10-variance-contributions","seed":202610106006,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"2/3\"],[\"-7\",\"2/9\"],[\"-11\",\"1/9\"]]"},"fields":{"title":"Every Centered Variance 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, 6] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[1250/243, 2888/729, 5476/729]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1250/243, 2888/729, 5476/729]."},"skills":["Every Centered Variance 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-r06-variance-0007","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afa4a33a46268eb70f86cdeafd7836354c99f07d36831203791f96374216705c","family_id":"qf-density10-variance-contributions","seed":202610106007,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"7/10\"],[\"-8\",\"1/5\"],[\"-13\",\"1/10\"]]"},"fields":{"title":"Every Centered Variance 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, 7] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[5887/1000, 2601/500, 10201/1000]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [5887/1000, 2601/500, 10201/1000]."},"skills":["Every Centered Variance 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-r06-variance-0008","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9386721467bbfdc251e9bbd0b50050ca381b06765bb4fbf0cd6f718d5bd9c2a7","family_id":"qf-density10-variance-contributions","seed":202610106008,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"8/11\"],[\"-9\",\"2/11\"],[\"-15\",\"1/11\"]]"},"fields":{"title":"Every Centered Variance 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, 8] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[72/11, 72/11, 144/11]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [72/11, 72/11, 144/11]."},"skills":["Every Centered Variance 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-r06-variance-0009","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07d2ead8dbd82df2ff4a4625db6d2ee6ec8e11673bae802f525eb5b467fe7640","family_id":"qf-density10-variance-contributions","seed":202610106009,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"3/4\"],[\"-10\",\"1/6\"],[\"-17\",\"1/12\"]]"},"fields":{"title":"Every Centered Variance 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, 9] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[1369/192, 6889/864, 27889/1728]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1369/192, 6889/864, 27889/1728]."},"skills":["Every Centered Variance 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-r06-variance-0010","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"61a366bb4e46e1e02493370852b034d4951e0cbbe79fdbf80c8421270853e63d","family_id":"qf-density10-variance-contributions","seed":202610106010,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"10/13\"],[\"-11\",\"2/13\"],[\"-19\",\"1/13\"]]"},"fields":{"title":"Every Centered Variance 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, 10] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[16810/2197, 20808/2197, 42436/2197]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [16810/2197, 20808/2197, 42436/2197]."},"skills":["Every Centered Variance 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-r06-variance-0011","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3513a3325b1cdf8fe2bf485453307df9c38eeac68d498107333bd6d4cbb3a8ab","family_id":"qf-density10-variance-contributions","seed":202610106011,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"11/14\"],[\"-12\",\"1/7\"],[\"-21\",\"1/14\"]]"},"fields":{"title":"Every Centered Variance 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, 11] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[22275/2744, 15129/1372, 62001/2744]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [22275/2744, 15129/1372, 62001/2744]."},"skills":["Every Centered Variance 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-r06-variance-0012","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3152c886435e10a7f672096fd654331dfa74a0a037a9aa92cb97484a742ecf1","family_id":"qf-density10-variance-contributions","seed":202610106012,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"4/5\"],[\"-13\",\"2/15\"],[\"-23\",\"1/15\"]]"},"fields":{"title":"Every Centered Variance 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, 12] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[9604/1125, 42632/3375, 87616/3375]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [9604/1125, 42632/3375, 87616/3375]."},"skills":["Every Centered Variance 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-r06-variance-0013","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40cccbf67410cd0217d1d6e206536c047cad019289397492999dfa1686e3eb84","family_id":"qf-density10-variance-contributions","seed":202610106013,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"13/16\"],[\"-14\",\"1/8\"],[\"-25\",\"1/16\"]]"},"fields":{"title":"Every Centered Variance 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, 13] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[36517/4096, 29241/2048, 120409/4096]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [36517/4096, 29241/2048, 120409/4096]."},"skills":["Every Centered Variance 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-r06-variance-0014","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"027b3caf3567b4c86e395b41cd708d990183da19bf35c268e2920f6f88103d98","family_id":"qf-density10-variance-contributions","seed":202610106014,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"14/17\"],[\"-15\",\"2/17\"],[\"-27\",\"1/17\"]]"},"fields":{"title":"Every Centered Variance 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, 14] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[45486/4913, 78408/4913, 161604/4913]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [45486/4913, 78408/4913, 161604/4913]."},"skills":["Every Centered Variance 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-r06-variance-0015","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01ff4c3c18ea9cef067b06e1f6cbed8c935332734610ca188602bc7f91e71de7","family_id":"qf-density10-variance-contributions","seed":202610106015,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"5/6\"],[\"-16\",\"1/9\"],[\"-29\",\"1/18\"]]"},"fields":{"title":"Every Centered Variance 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, 15] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[18605/1944, 51529/2916, 212521/5832]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [18605/1944, 51529/2916, 212521/5832]."},"skills":["Every Centered Variance 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-r06-variance-0016","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8e44d3e23eed147b3fff3cdd8042c240abfa6b2830bb231a6c27c428eeddaa5a","family_id":"qf-density10-variance-contributions","seed":202610106016,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"16/19\"],[\"-17\",\"2/19\"],[\"-31\",\"1/19\"]]"},"fields":{"title":"Every Centered Variance 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, 16] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[67600/6859, 133128/6859, 274576/6859]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [67600/6859, 133128/6859, 274576/6859]."},"skills":["Every Centered Variance 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-r06-variance-0017","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"06d4ef2adedb653dedbd18c833eedabeaa2206facbdbd779b94ba07db07730ef","family_id":"qf-density10-variance-contributions","seed":202610106017,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"17/20\"],[\"-18\",\"1/10\"],[\"-33\",\"1/20\"]]"},"fields":{"title":"Every Centered Variance 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, 17] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[80937/8000, 84681/4000, 349281/8000]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [80937/8000, 84681/4000, 349281/8000]."},"skills":["Every Centered Variance 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-r06-variance-0018","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4a811ff3867349c149acdeb7c45599d9b08d2355234a38243e250d511002c5","family_id":"qf-density10-variance-contributions","seed":202610106018,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"6/7\"],[\"-19\",\"2/21\"],[\"-35\",\"1/21\"]]"},"fields":{"title":"Every Centered Variance 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, 18] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[10658/1029, 212552/9261, 438244/9261]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [10658/1029, 212552/9261, 438244/9261]."},"skills":["Every Centered Variance 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-r06-variance-0019","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1aeda02685bc7e6a41e7d4e19d1e92e984db7fecfbcf9f14b3976be358eea58","family_id":"qf-density10-variance-contributions","seed":202610106019,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"19/22\"],[\"-20\",\"1/11\"],[\"-37\",\"1/22\"]]"},"fields":{"title":"Every Centered Variance 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, 19] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[931/88, 99/4, 4489/88]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [931/88, 99/4, 4489/88]."},"skills":["Every Centered Variance 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-r06-variance-0020","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a6246f32b59adf5541e431257f35d715baddc5001a260f94624165a6487cadce","family_id":"qf-density10-variance-contributions","seed":202610106020,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"20/23\"],[\"-21\",\"2/23\"],[\"-39\",\"1/23\"]]"},"fields":{"title":"Every Centered Variance 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, 20] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[131220/12167, 323208/12167, 665856/12167]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [131220/12167, 323208/12167, 665856/12167]."},"skills":["Every Centered Variance 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-r06-variance-0021","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"131bdc31fbee4f404b0b19e4a3079e0719f430ae08d3b51f5d515e582367f5fc","family_id":"qf-density10-variance-contributions","seed":202610106021,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"7/8\"],[\"-22\",\"1/12\"],[\"-41\",\"1/24\"]]"},"fields":{"title":"Every Centered Variance 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, 21] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[50575/4608, 196249/6912, 808201/13824]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [50575/4608, 196249/6912, 808201/13824]."},"skills":["Every Centered Variance 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-r06-variance-0022","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"09b3bb312d5287aa493b30a346255471cef1b5913708f6f1defece1a705ae2ff","family_id":"qf-density10-variance-contributions","seed":202610106022,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"22/25\"],[\"-23\",\"2/25\"],[\"-43\",\"1/25\"]]"},"fields":{"title":"Every Centered Variance 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, 22] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[174262/15625, 472392/15625, 972196/15625]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [174262/15625, 472392/15625, 972196/15625]."},"skills":["Every Centered Variance 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-r06-variance-0023","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d87771c1b07dd2da3ff5441aad4d2aba6d90ff6e555b3d71adc5d718181c3b24","family_id":"qf-density10-variance-contributions","seed":202610106023,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"23/26\"],[\"-24\",\"1/13\"],[\"-45\",\"1/26\"]]"},"fields":{"title":"Every Centered Variance 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, 23] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[198927/17576, 281961/8788, 1159929/17576]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [198927/17576, 281961/8788, 1159929/17576]."},"skills":["Every Centered Variance 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-r06-variance-0024","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4c615674caca13f2e9f1c56c88cfd2859cb076ddc8dc00dfe5761d7d47e29f2f","family_id":"qf-density10-variance-contributions","seed":202610106024,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"8/9\"],[\"-25\",\"2/27\"],[\"-47\",\"1/27\"]]"},"fields":{"title":"Every Centered Variance 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, 24] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[75272/6561, 668168/19683, 1373584/19683]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [75272/6561, 668168/19683, 1373584/19683]."},"skills":["Every Centered Variance 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-r06-variance-0025","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a1e4a86b3cdaa3c93f0470092f29927db1dcb1f2b7c451dbfa63a3d623fe645","family_id":"qf-density10-variance-contributions","seed":202610106025,"semantic_template":"qf-density10-variance-contributions/[[\"0\",\"25/28\"],[\"-26\",\"1/14\"],[\"-49\",\"1/28\"]]"},"fields":{"title":"Every Centered Variance 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, 25] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"[255025/21952, 393129/10976, 1615441/21952]","explanation":"Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [255025/21952, 393129/10976, 1615441/21952]."},"skills":["Every Centered Variance 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-r06-variance-0026","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"588f36f61ff2757a57729a52084b49376f2d887f8d328cd0d0081a6e6ec07d1a","family_id":"qf-density10-variance-second-moment-pair","seed":202610106026,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-4\",\"1/1\"]]"},"fields":{"title":"Second Moment and Squared 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=[4, 99, 4] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(16/1, 16/1)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (16/1, 16/1)."},"skills":["Second Moment and Squared 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-r06-variance-0027","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4e1fbfbe43f0aab123ce0997b0351a88fbce117e5fe8092a04ad1f2a3b927e24","family_id":"qf-density10-variance-second-moment-pair","seed":202610106027,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-1\",\"1/2\"],[\"1\",\"1/2\"]]"},"fields":{"title":"Second Moment and Squared Mean","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=[-1, 1] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1/1, 0/1)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (1/1, 0/1)."},"skills":["Second Moment and Squared 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-r06-variance-0028","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"90eb51041fd18d11ef0de8cfed9aa4ebff25a55eb4dfc7af5b406fda0c2c771f","family_id":"qf-density10-variance-second-moment-pair","seed":202610106028,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/2\"],[\"2\",\"1/3\"],[\"3\",\"1/6\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 3] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(29/6, 1/36)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (29/6, 1/36)."},"skills":["Second Moment and Squared 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-r06-variance-0029","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17a2dd353a7874876b3280f85ca0ad87ca54fc584acfd16a070b799d3b168040","family_id":"qf-density10-variance-second-moment-pair","seed":202610106029,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/7\"],[\"-4\",\"1/7\"],[\"3\",\"4/7\"]]"},"fields":{"title":"Second Moment and Squared 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=[-3, 2, 4] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(60/7, 16/49)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (60/7, 16/49)."},"skills":["Second Moment and Squared 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-r06-variance-0030","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d1111d94abeeba882fbbf26226d1a560a3cb9fbcab6be87a35969906721d48df","family_id":"qf-density10-variance-second-moment-pair","seed":202610106030,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/4\"],[\"-5\",\"1/8\"],[\"4\",\"5/8\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 5] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(113/8, 121/64)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (113/8, 121/64)."},"skills":["Second Moment and Squared 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-r06-variance-0031","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ced1ff2f3a41f797adc22b3f5e02183bdfe4f1830f0d50b979220ee14acafd0","family_id":"qf-density10-variance-second-moment-pair","seed":202610106031,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/9\"],[\"-6\",\"1/9\"],[\"5\",\"2/3\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 6] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(194/9, 400/81)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (194/9, 400/81)."},"skills":["Second Moment and Squared 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-r06-variance-0032","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dbdce2495f1d8832ee3c0d1ea761d9b04cc10f9e5711271e81027ec8570b98ff","family_id":"qf-density10-variance-second-moment-pair","seed":202610106032,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/5\"],[\"-7\",\"1/10\"],[\"6\",\"7/10\"]]"},"fields":{"title":"Second Moment and Squared 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=[-6, 2, 7] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(309/10, 961/100)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (309/10, 961/100)."},"skills":["Second Moment and Squared 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-r06-variance-0033","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71858febec4548f37e1cfb1bb163859f033f6341344440d275a83d75fdbfd460","family_id":"qf-density10-variance-second-moment-pair","seed":202610106033,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/11\"],[\"-8\",\"1/11\"],[\"7\",\"8/11\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 8] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(464/11, 16/1)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (464/11, 16/1)."},"skills":["Second Moment and Squared 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-r06-variance-0034","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e1a5c9e31cfaeee85a46e8ef95fa66cd1798f0637e6f6bb85b168481aa86dea6","family_id":"qf-density10-variance-second-moment-pair","seed":202610106034,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/6\"],[\"-9\",\"1/12\"],[\"8\",\"3/4\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 9] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(665/12, 3481/144)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (665/12, 3481/144)."},"skills":["Second Moment and Squared 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-r06-variance-0035","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5e51bf89e636fa181687abcb84155c9cce29ab5f1ed1c3b0eb9c37a0e528e30","family_id":"qf-density10-variance-second-moment-pair","seed":202610106035,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-10\",\"1/13\"],[\"-2\",\"2/13\"],[\"9\",\"10/13\"]]"},"fields":{"title":"Second Moment and Squared 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=[-9, 2, 10] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(918/13, 5776/169)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (918/13, 5776/169)."},"skills":["Second Moment and Squared 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-r06-variance-0036","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2538f3f2d84f456082a26681ce4663ce5f48fcbca41f8651efdb4f8fe4ee3734","family_id":"qf-density10-variance-second-moment-pair","seed":202610106036,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-10\",\"11/14\"],[\"11\",\"1/14\"],[\"2\",\"1/7\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 11] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1229/14, 9025/196)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (1229/14, 9025/196)."},"skills":["Second Moment and Squared 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-r06-variance-0037","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8ea5f25f2af655fe18fb144c1b43b4a060dd54fdfeb48445c1e2320dfcd7605","family_id":"qf-density10-variance-second-moment-pair","seed":202610106037,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-11\",\"4/5\"],[\"12\",\"1/15\"],[\"2\",\"2/15\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 12] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(1604/15, 13456/225)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (1604/15, 13456/225)."},"skills":["Second Moment and Squared 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-r06-variance-0038","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a2ccf5bd79a62f8a66697b9609f1cccc555adc105ca101b6d41b75414eccc66","family_id":"qf-density10-variance-second-moment-pair","seed":202610106038,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-12\",\"13/16\"],[\"13\",\"1/16\"],[\"2\",\"1/8\"]]"},"fields":{"title":"Second Moment and Squared 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=[-12, 2, 13] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2049/16, 19321/256)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (2049/16, 19321/256)."},"skills":["Second Moment and Squared 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-r06-variance-0039","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"11b6f4483b851fb5d2f3d3088f7fb62b245ff4d9551e41dda4a1d7f5fb73e55f","family_id":"qf-density10-variance-second-moment-pair","seed":202610106039,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-13\",\"14/17\"],[\"14\",\"1/17\"],[\"2\",\"2/17\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 14] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(2570/17, 26896/289)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (2570/17, 26896/289)."},"skills":["Second Moment and Squared 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-r06-variance-0040","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c447539112733fe063ae89c0f3d3c232e000d85d4c68f99e087e250cb8617dc","family_id":"qf-density10-variance-second-moment-pair","seed":202610106040,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-14\",\"5/6\"],[\"15\",\"1/18\"],[\"2\",\"1/9\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 15] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3173/18, 36481/324)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (3173/18, 36481/324)."},"skills":["Second Moment and Squared 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-r06-variance-0041","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6041c233e8fdf53471da808f4aad5ef88dc8405c822992d14f374554076bd19f","family_id":"qf-density10-variance-second-moment-pair","seed":202610106041,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-15\",\"16/19\"],[\"16\",\"1/19\"],[\"2\",\"2/19\"]]"},"fields":{"title":"Second Moment and Squared 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=[-15, 2, 16] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(3864/19, 48400/361)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (3864/19, 48400/361)."},"skills":["Second Moment and Squared 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-r06-variance-0042","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b1ee1f9a036320ed907afd20102c0f7d548c26fda458379d988a2e9871c94bf4","family_id":"qf-density10-variance-second-moment-pair","seed":202610106042,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-16\",\"17/20\"],[\"17\",\"1/20\"],[\"2\",\"1/10\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 17] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(4649/20, 63001/400)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (4649/20, 63001/400)."},"skills":["Second Moment and Squared 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-r06-variance-0043","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d028994104ba000b924011416d56022c741d662541797897c262d14ec75ba4c","family_id":"qf-density10-variance-second-moment-pair","seed":202610106043,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-17\",\"6/7\"],[\"18\",\"1/21\"],[\"2\",\"2/21\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 18] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(5534/21, 80656/441)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (5534/21, 80656/441)."},"skills":["Second Moment and Squared 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-r06-variance-0044","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e6ac4b7a7bc0e885fbe7fa84c573adba6443cc39390f582c568a8aa61e1c7d5","family_id":"qf-density10-variance-second-moment-pair","seed":202610106044,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-18\",\"19/22\"],[\"19\",\"1/22\"],[\"2\",\"1/11\"]]"},"fields":{"title":"Second Moment and Squared 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=[-18, 2, 19] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(6525/22, 841/4)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (6525/22, 841/4)."},"skills":["Second Moment and Squared 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-r06-variance-0045","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d521167b0fc488633afee267593ae43551f001445481433397deb250164aa15b","family_id":"qf-density10-variance-second-moment-pair","seed":202610106045,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-19\",\"20/23\"],[\"2\",\"2/23\"],[\"20\",\"1/23\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 20] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(7628/23, 126736/529)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (7628/23, 126736/529)."},"skills":["Second Moment and Squared 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-r06-variance-0046","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5468d6042ca6b489f4a8abb670ee98aaf5bf1b737b9be004c824239772476b88","family_id":"qf-density10-variance-second-moment-pair","seed":202610106046,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/12\"],[\"-21\",\"1/24\"],[\"20\",\"7/8\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 21] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(8849/24, 156025/576)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (8849/24, 156025/576)."},"skills":["Second Moment and Squared 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-r06-variance-0047","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ede59c656089edf9254187c7e2116a4004c9f9ec694c0bf948ddcca22e9041a8","family_id":"qf-density10-variance-second-moment-pair","seed":202610106047,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/25\"],[\"-22\",\"1/25\"],[\"21\",\"22/25\"]]"},"fields":{"title":"Second Moment and Squared 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=[-21, 2, 22] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(10194/25, 190096/625)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (10194/25, 190096/625)."},"skills":["Second Moment and Squared 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-r06-variance-0048","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70fbe8c117b08c7a8e06c54236ee317c3240e57f29c30cf62e6fca91af131309","family_id":"qf-density10-variance-second-moment-pair","seed":202610106048,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/13\"],[\"-23\",\"1/26\"],[\"22\",\"23/26\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 23] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(11669/26, 229441/676)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (11669/26, 229441/676)."},"skills":["Second Moment and Squared 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-r06-variance-0049","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"35d104fb38d908a5bf3f7a42ed7084b1b12835722d6b5871a527dcc486931efa","family_id":"qf-density10-variance-second-moment-pair","seed":202610106049,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"2/27\"],[\"-24\",\"1/27\"],[\"23\",\"8/9\"]]"},"fields":{"title":"Second Moment and Squared 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, 2, 24] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(13280/27, 274576/729)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (13280/27, 274576/729)."},"skills":["Second Moment and Squared 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-r06-variance-0050","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70ce86bbb721d416c4f64e9132d7497b0fc4e3be860a962805fc29c6ec1bfc86","family_id":"qf-density10-variance-second-moment-pair","seed":202610106050,"semantic_template":"qf-density10-variance-second-moment-pair/[[\"-2\",\"1/14\"],[\"-25\",\"1/28\"],[\"24\",\"25/28\"]]"},"fields":{"title":"Second Moment and Squared 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=[-24, 2, 25] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"(15033/28, 326041/784)","explanation":"Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (15033/28, 326041/784)."},"skills":["Second Moment and Squared 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-r06-variance-0051","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"29cf05b1030d8e21bc3c311158bb402d3309c1a60f4d29bff5779c0d24894f5d","family_id":"qf-density10-variance-difference","seed":202610106051,"semantic_template":"qf-density10-variance-difference/[[\"0\",\"1/1\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"0/1","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 0/1."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0052","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85a8eb205f9fbd609790cc2b1dee59c458352abb72da76f5cc21ffb7bdeefab0","family_id":"qf-density10-variance-difference","seed":202610106052,"semantic_template":"qf-density10-variance-difference/[[\"-4\",\"1/2\"],[\"0\",\"1/2\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"4/1","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4/1."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0053","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"279d0d6e170994f8f75ebb5ad9691b211ba4758de596cb18c1fec2f0bcc80ce4","family_id":"qf-density10-variance-difference","seed":202610106053,"semantic_template":"qf-density10-variance-difference/[[\"-6\",\"1/3\"],[\"-8\",\"1/6\"],[\"0\",\"1/2\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"104/9","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 104/9."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0054","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d8a328ddae5ab02cd766c18cc421841130010222acf2c579811a8ef46320107","family_id":"qf-density10-variance-difference","seed":202610106054,"semantic_template":"qf-density10-variance-difference/[[\"-11\",\"1/7\"],[\"-7\",\"2/7\"],[\"0\",\"4/7\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"908/49","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 908/49."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0055","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7384b2ed944de87396cfe82267131c44aca6a9f0dead280ddf122f1400e54d5","family_id":"qf-density10-variance-difference","seed":202610106055,"semantic_template":"qf-density10-variance-difference/[[\"-14\",\"1/8\"],[\"-8\",\"1/4\"],[\"0\",\"5/8\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"423/16","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 423/16."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0056","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fec91caef45b03086f1753b1d75bafec09b6c5541d047afd68f7d78d51edd991","family_id":"qf-density10-variance-difference","seed":202610106056,"semantic_template":"qf-density10-variance-difference/[[\"-17\",\"1/9\"],[\"-9\",\"2/9\"],[\"0\",\"2/3\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"2834/81","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2834/81."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0057","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42dd0a1b177b3ed0ca6423a12d7c65823c5de37f5912775b655379cd1c0b4e18","family_id":"qf-density10-variance-difference","seed":202610106057,"semantic_template":"qf-density10-variance-difference/[[\"-10\",\"1/5\"],[\"-20\",\"1/10\"],[\"0\",\"7/10\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"44/1","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 44/1."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0058","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dfee6a4377fef70e965d122858057546416b8fc9fb4721cd79379ecbadfe4d2","family_id":"qf-density10-variance-difference","seed":202610106058,"semantic_template":"qf-density10-variance-difference/[[\"-11\",\"2/11\"],[\"-23\",\"1/11\"],[\"0\",\"8/11\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"6456/121","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6456/121."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0059","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4577d8bcab7a0664452c54d1ad4233f5bfe7f04d0908e5cb6eaa390df2b1b81f","family_id":"qf-density10-variance-difference","seed":202610106059,"semantic_template":"qf-density10-variance-difference/[[\"-12\",\"1/6\"],[\"-26\",\"1/12\"],[\"0\",\"3/4\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"2267/36","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2267/36."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0060","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a71b24016e492df4ece610c87b7319d4d7457b18f20fc13001566fa9b570b6b7","family_id":"qf-density10-variance-difference","seed":202610106060,"semantic_template":"qf-density10-variance-difference/[[\"-13\",\"2/13\"],[\"-29\",\"1/13\"],[\"0\",\"10/13\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"12302/169","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 12302/169."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0061","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d67975e41d4ed2cc445b13e48c000d34bbbf477d336efb923374ac713560055d","family_id":"qf-density10-variance-difference","seed":202610106061,"semantic_template":"qf-density10-variance-difference/[[\"-14\",\"1/7\"],[\"-32\",\"1/14\"],[\"0\",\"11/14\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"4056/49","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4056/49."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0062","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f71387de06dcc987a362603d6ffb62b12a5ec372706017c7fc6b069b86fa1692","family_id":"qf-density10-variance-difference","seed":202610106062,"semantic_template":"qf-density10-variance-difference/[[\"-15\",\"2/15\"],[\"-35\",\"1/15\"],[\"0\",\"4/5\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"836/9","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 836/9."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0063","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1e4c95eccb4966d799ada76494aca07e9b51dd866244f03605c7f200e08b1021","family_id":"qf-density10-variance-difference","seed":202610106063,"semantic_template":"qf-density10-variance-difference/[[\"-16\",\"1/8\"],[\"-38\",\"1/16\"],[\"0\",\"13/16\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"6599/64","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6599/64."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0064","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9a04e82fdacbc9cc10812ffdabdf70ba0dfef60e1ec7cdb946e473bbe83a9b60","family_id":"qf-density10-variance-difference","seed":202610106064,"semantic_template":"qf-density10-variance-difference/[[\"-17\",\"2/17\"],[\"-41\",\"1/17\"],[\"0\",\"14/17\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"32778/289","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 32778/289."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0065","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7609ffc3948d2a3800c27d2aaf36ae98e4a9952962c6ade2bac18fe5f8db46c7","family_id":"qf-density10-variance-difference","seed":202610106065,"semantic_template":"qf-density10-variance-difference/[[\"-18\",\"1/9\"],[\"-44\",\"1/18\"],[\"0\",\"5/6\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"10028/81","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 10028/81."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0066","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3bad2b581eb2310eaf4be48dbb4550306b8c31cdf6dd9ca1d1ff7705f1774bbc","family_id":"qf-density10-variance-difference","seed":202610106066,"semantic_template":"qf-density10-variance-difference/[[\"-19\",\"2/19\"],[\"-47\",\"1/19\"],[\"0\",\"16/19\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"48464/361","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 48464/361."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0067","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bd5e0c879980a22ca4c0bc9cd171e49e7d3fdc1b26b59093646dc04a48cfd03b","family_id":"qf-density10-variance-difference","seed":202610106067,"semantic_template":"qf-density10-variance-difference/[[\"-20\",\"1/10\"],[\"-50\",\"1/20\"],[\"0\",\"17/20\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"579/4","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 579/4."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0068","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7ae7285a0710641759d8c8950f1866759bda602568716e8f1a15d69654c44e74","family_id":"qf-density10-variance-difference","seed":202610106068,"semantic_template":"qf-density10-variance-difference/[[\"-21\",\"2/21\"],[\"-53\",\"1/21\"],[\"0\",\"6/7\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"68486/441","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 68486/441."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0069","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2861b5bb0e655c62fcd9d3d7483a9cf3954a816c8836bed34fedb81f5b4da01d","family_id":"qf-density10-variance-difference","seed":202610106069,"semantic_template":"qf-density10-variance-difference/[[\"-22\",\"1/11\"],[\"-56\",\"1/22\"],[\"0\",\"19/22\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"20072/121","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 20072/121."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0070","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfb0dda4f226a7fe488d5fabb1066d8fd3406518d9b8378d0dab5fae4bd1338a","family_id":"qf-density10-variance-difference","seed":202610106070,"semantic_template":"qf-density10-variance-difference/[[\"-23\",\"2/23\"],[\"-59\",\"1/23\"],[\"0\",\"20/23\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"93372/529","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 93372/529."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0071","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a1918cfd733210cfa430145e95b09551046ec509eaad97e7fefad29b4adb780c","family_id":"qf-density10-variance-difference","seed":202610106071,"semantic_template":"qf-density10-variance-difference/[[\"-24\",\"1/12\"],[\"-62\",\"1/24\"],[\"0\",\"7/8\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"26951/144","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 26951/144."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0072","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"794abb9d9ad7c3a4e82671df91eb0146d47f46083ebf8ab413cafe488af9aeb0","family_id":"qf-density10-variance-difference","seed":202610106072,"semantic_template":"qf-density10-variance-difference/[[\"-25\",\"2/25\"],[\"-65\",\"1/25\"],[\"0\",\"22/25\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"4946/25","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4946/25."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0073","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c49bd580fa83c523cff715547d277d20dcd1cb0bfa14730fc5a8e55d3c5dd52","family_id":"qf-density10-variance-difference","seed":202610106073,"semantic_template":"qf-density10-variance-difference/[[\"-26\",\"1/13\"],[\"-68\",\"1/26\"],[\"0\",\"23/26\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"35244/169","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 35244/169."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0074","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"129b7003361988285c79e77c6906ff57a478f23e139efd9519351c2d6e7cccb8","family_id":"qf-density10-variance-difference","seed":202610106074,"semantic_template":"qf-density10-variance-difference/[[\"-27\",\"2/27\"],[\"-71\",\"1/27\"],[\"0\",\"8/9\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"159848/729","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 159848/729."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0075","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"46bebd2c625a38d9d7336e62f0ab7cc84aa04de28effd32661d2b0b0ee131deb","family_id":"qf-density10-variance-difference","seed":202610106075,"semantic_template":"qf-density10-variance-difference/[[\"-28\",\"1/14\"],[\"-74\",\"1/28\"],[\"0\",\"25/28\"]]"},"fields":{"title":"Variance of a Shared-Model Difference","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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.","answer":{"kind":"expression","value":"45083/196","explanation":"Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 45083/196."},"skills":["Variance of a Shared-Model Difference"],"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-r06-variance-0076","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"783c3b2ea50650657b2cee5e83e83dc8f742504fb71742d301d43edce4003fac","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106076,"semantic_template":"qf-density10-variance-first-support-mismatch/[\"constant_positive_support\"]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 2, 0, 1, 0, 0, 0, 0], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[99, 4, -99, 4, 9, 8, 7, 6] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"none","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is none."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0077","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e4701e0a895c6a2a3085c06e9a0f2715026efbd2df0b903a8d7c0ec56bacfde","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106077,"semantic_template":"qf-density10-variance-first-support-mismatch/[0,1]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[2, 2, 0, 0, 0, 0, 0, 0], D=4, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 2, 102, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"1","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 1."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0078","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5fb736ef800dc560962d92eff118a576705005589a4a041aee38596854524244","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106078,"semantic_template":"qf-density10-variance-first-support-mismatch/[1,2]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 3, 2, 0, 0, 0, 0, 0], D=5, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 3, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"2","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0079","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a7af9710c5e46de41029ad5e349499b765d7eeda48541863623d4189f4f97a1","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106079,"semantic_template":"qf-density10-variance-first-support-mismatch/[2,3]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 4, 2, 0, 0, 0, 0], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 4, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"3","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 3."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0080","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf91b50b68fc1b94fb7429588b93d39beb77ab006c01b77880dac6defc81ad50","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106080,"semantic_template":"qf-density10-variance-first-support-mismatch/[3,4]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 5, 2, 0, 0, 0], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 5, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"4","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0081","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"451daf988b53ba3db65913f3d067e7fcd20ca8c6fc49f134fa3fc9172f8ac097","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106081,"semantic_template":"qf-density10-variance-first-support-mismatch/[4,5]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 6, 2, 0, 0], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 0, 6, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"5","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0082","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"14049de42efefe0da939180091652cf3c4c161c9143ebee538532cc2a931cefb","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106082,"semantic_template":"qf-density10-variance-first-support-mismatch/[5,6]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 0, 7, 2, 0], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 104, 0, 7, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"6","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0083","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8503b3461d660b83a0b24c59c9db268da88bccee348bc26309516ed0f504954a","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106083,"semantic_template":"qf-density10-variance-first-support-mismatch/[6,7]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 0, 0, 8, 2], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 104, 105, 0, 8] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"7","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0084","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7fc6b705d8221ac0ec921bc79a2a27e65eb9936f49d095139975d8bd5852194e","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106084,"semantic_template":"qf-density10-variance-first-support-mismatch/[0,2]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[9, 0, 2, 0, 0, 0, 0, 0], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 9, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"2","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0085","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1dde4f6847c3c05c116643fa158bd3a4ce4948522474f84c08ba82bd1ab44ea9","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106085,"semantic_template":"qf-density10-variance-first-support-mismatch/[1,3]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 10, 0, 2, 0, 0, 0, 0], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 10, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"3","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 3."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0086","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86212378f3b61975ed3b5b61160d30e35e6e95853c5eae8e7eb624fefbc580eb","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106086,"semantic_template":"qf-density10-variance-first-support-mismatch/[2,4]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 11, 0, 2, 0, 0, 0], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 103, 11, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"4","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0087","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0440e37a95997de7a54ab50eee519c23c86cc32d49b4ca2cb032ce786c6e1d","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106087,"semantic_template":"qf-density10-variance-first-support-mismatch/[3,5]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 12, 0, 2, 0, 0], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 12, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"5","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0088","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"caa08a0be5d7702a1ea1af1f05e8d7696b98b60aee2a0b8836dfe161e8a575c7","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106088,"semantic_template":"qf-density10-variance-first-support-mismatch/[4,6]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 13, 0, 2, 0], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 0, 105, 13, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"6","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0089","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bac780b038ec308d87c196543fc36f74452e27d2ee5991e17c1567c9901dcaf6","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106089,"semantic_template":"qf-density10-variance-first-support-mismatch/[5,7]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 0, 14, 0, 2], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 104, 0, 106, 14] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"7","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0090","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad12063a48c46071d04202dbc7d82bbfedce47d086f0ab32d3b700d8a81ccc3f","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106090,"semantic_template":"qf-density10-variance-first-support-mismatch/[0,3]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[15, 0, 0, 2, 0, 0, 0, 0], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 102, 15, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"3","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 3."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0091","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2722aa0e6c96006668e95883a20eac86683617e8e64e617f67f6ef701e23e74","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106091,"semantic_template":"qf-density10-variance-first-support-mismatch/[1,4]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 16, 0, 0, 2, 0, 0, 0], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 103, 16, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"4","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0092","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afe2ecb8b978ca98b07f47c51625572abbcf8933b2467c9b6156420826cd2d9b","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106092,"semantic_template":"qf-density10-variance-first-support-mismatch/[2,5]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 17, 0, 0, 2, 0, 0], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 103, 104, 17, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"5","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0093","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"98cf3d0042c0d99377bc57ef13d2e4d0ad70c5a14ffc4a403432637ffe81fea2","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106093,"semantic_template":"qf-density10-variance-first-support-mismatch/[3,6]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 18, 0, 0, 2, 0], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 105, 18, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"6","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0094","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1d700d42a4724664354c444a9a672f6f5f7e1b6f5956cc17a466acf92e54f7ea","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106094,"semantic_template":"qf-density10-variance-first-support-mismatch/[4,7]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 19, 0, 0, 2], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 0, 105, 106, 19] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"7","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0095","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c274f610f55530a51e1dada2bcd0be10bdefe96a255c6e395def78422feff8cc","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106095,"semantic_template":"qf-density10-variance-first-support-mismatch/[0,4]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[20, 0, 0, 0, 2, 0, 0, 0], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 102, 103, 20, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"4","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0096","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8dc54e87bd44cefd27ffef6b6282046f54a8ef8f9f64f75436ac1f7eb54ec060","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106096,"semantic_template":"qf-density10-variance-first-support-mismatch/[1,5]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 21, 0, 0, 0, 2, 0, 0], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 103, 104, 21, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"5","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0097","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc84d692f86b988c49319318998f30d02815e677cd8e2f91b7c77201d5e987c3","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106097,"semantic_template":"qf-density10-variance-first-support-mismatch/[2,6]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 22, 0, 0, 0, 2, 0], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 103, 104, 105, 22, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"6","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0098","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c3bb06c3471984e924c3dcf3263f396550e3db416eef2db40b930261f7fe058f","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106098,"semantic_template":"qf-density10-variance-first-support-mismatch/[3,7]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 0, 0, 23, 0, 0, 0, 2], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 105, 106, 23] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"7","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0099","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a768e5a847dbd0392ba95e85a304ccbc74bf6d657406232ae359c3e28dd1a9c","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106099,"semantic_template":"qf-density10-variance-first-support-mismatch/[0,5]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[24, 0, 0, 0, 0, 2, 0, 0], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 102, 103, 104, 24, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"5","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5."},"skills":["First Mismatch on Positive Probability Support"],"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-r06-variance-0100","batch_id":"batch-20261008-advance-20261010-r06-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e03d92d8bd705cf30704397ae65ddff630c23a69edb407692284ad392b19097","family_id":"qf-density10-variance-first-support-mismatch","seed":202610106100,"semantic_template":"qf-density10-variance-first-support-mismatch/[1,6]"},"fields":{"title":"First Mismatch on Positive Probability Support","stem":"S={0,…,7}, F=P(S), masses=[0, 25, 0, 0, 0, 0, 2, 0], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 103, 104, 105, 25, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.","answer":{"kind":"short_text","value":"6","explanation":"Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6."},"skills":["First Mismatch on Positive Probability Support"],"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."}}