{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0001","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf00d6bc8f786e1ec2b72bdddea427a5e45eb45ddea77a8cb784be2472e3dc44","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227001,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,4]],\"outcomes\":[0,1,3]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([0, 1, 3], [1/4, 1/4, 1/4], Yes, 3/4, No, No)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/4], Yes, 3/4, No, No)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0002","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"063174742d6304515a96cb169f9cf772ddbdc741456f4d2cfde285a221126a6c","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227002,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[1,2,4]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([1, 2, 4], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0003","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85e9f48f914a240ef1f44397126caee5ffdb762a1d82a58b33a6d23e0d45f43f","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227003,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[2,3,5]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([2, 3, 5], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0004","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"18e5f2c77345b03f3ac1546aac4d4e4eea166e10c76584b8245d8df9273f0887","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227004,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[3,4,6]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([3, 4, 6], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0005","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f54d6bd411d7df72d92add14efa3ba04b03cc03debe5dddd4bc1e9f44d125991","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227005,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[4,5,7]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([4, 5, 7], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0006","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a59342836153a8dd215103fb58e10da0b848171c69a719feaac57e6d2f1025e7","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227006,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,4]],\"outcomes\":[5,6,8]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([5, 6, 8], [1/4, 1/4, 1/4], Yes, 3/4, No, No)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/4], Yes, 3/4, No, No)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0007","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"511a63fcf686b69b673fb884f0f9c663bedd21774b3ea69f2c485b864f87e253","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227007,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[6,7,9]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([6, 7, 9], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0008","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7cef5835ccf8dcac8013c624c76c965ff137c1711b739b55ea052b66ddf33201","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227008,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[7,8,10]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([7, 8, 10], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0009","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f919f834fbbd7460bad62c4bd52301fb1219500f9f4ae452d2b410c85049e09","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227009,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[8,9,11]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([8, 9, 11], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0010","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"071fa853948b23439eccfb7b2f577713b6ea5003ec5ad74036f9b1c8d460f577","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227010,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[9,10,12]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([9, 10, 12], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0011","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aae924979fe95974658982f527b526b971f62a878abb49ae33ca58ff2dd69107","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227011,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,4]],\"outcomes\":[10,11,13]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([10, 11, 13], [1/4, 1/4, 1/4], Yes, 3/4, No, No)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/4], Yes, 3/4, No, No)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0012","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e480344edfcda30ee1c142746a47f63575f9ee9b47cb9b75aaf7f3c57655bd5","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227012,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[11,12,14]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([11, 12, 14], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0013","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"280d91121c274867c977ef005f8e55feb0d4028560571de634554b160781229b","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227013,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[12,13,15]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([12, 13, 15], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0014","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b9965759618386f1760f04d1b1e42d4f41679d2cf766da1ee4e671b9b332e131","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227014,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[13,14,16]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([13, 14, 16], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0015","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7a20ca3d29acdc522fb1c8e7d5a5f028ef180d2d5990925f94e58df1b084878","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227015,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[14,15,17]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([14, 15, 17], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0016","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ba04f9fbff51fd12acd483cb491e01aa5d2b58186cd4e42698a17c7745e9535","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227016,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,4]],\"outcomes\":[15,16,18]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([15, 16, 18], [1/4, 1/4, 1/4], Yes, 3/4, No, No)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/4], Yes, 3/4, No, No)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0017","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef2cbadfbd2f96579da468eba6a04abab40983981192859d68ea0a3843dd1da2","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227017,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[16,17,19]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([16, 17, 19], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0018","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7b9af28792f7a62a5aec83395bb4d5b1d91f48635f6882e126859cf05d6e08e","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227018,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[17,18,20]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([17, 18, 20], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0019","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f2b4987d6ce35fdda1a3221ccd6ecd02f1e52b7d9737e918aff77192cedd453","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227019,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[18,19,21]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([18, 19, 21], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0020","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1500178f961e38697944ebaed4df5c28d1296a0314f50f023128e9ef5fb1bb71","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227020,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[19,20,22]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([19, 20, 22], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0021","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4deaefa6288f946714ac0532e3154a6ba8281348c09f3f8588ec547bdee0be99","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227021,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,4]],\"outcomes\":[20,21,23]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([20, 21, 23], [1/4, 1/4, 1/4], Yes, 3/4, No, No)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/4], Yes, 3/4, No, No)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0022","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b4c6e4225eab59bf3daea59a7e868c7e3c27b49a5d5ffcbc157276876daab9cf","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227022,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[21,22,24]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([21, 22, 24], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0023","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a997be77059ee384f6ffbed98b4fa1fe27b5da2bd6c23a6cd074efcd16962bce","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227023,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[22,23,25]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([22, 23, 25], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0024","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b254e8a02b531729895630c1e78d7d044c6d966131c2239bced95f654653208a","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227024,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[23,24,26]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([23, 24, 26], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0025","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4adfc94f0a56f05d9e82855a2adcf35b541c4c9c40c53dc7ef8817e1d24d9da4","family_id":"qf-str22-distribution-pmf-qualification","seed":202610227025,"semantic_template":"qf-str22-distribution-pmf-qualification/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[24,25,27]}"},"fields":{"title":"Exact Normalization and Nonnegativity before PMF Use","stem":"Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).","answer":{"kind":"expression","value":"([24, 25, 27], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)","explanation":"Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)."},"skills":["Exact Normalization and Nonnegativity before PMF Use"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0026","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3ecb2d1b9b3cf7b50ff631403ae94325a4bef0002e584f4a24bbd790c1583be","family_id":"qf-str22-distribution-event-probability","seed":202610227026,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[0,3],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[0,1,3]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[0, 3] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([0, 1, 3], [1/4, 1/4, 1/2], [0, 3], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/2], [0, 3], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0027","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"85b6b77b29db924b7a939aede3cb7f0285766acdb36ac6488611496f034a6f50","family_id":"qf-str22-distribution-event-probability","seed":202610227027,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[1,4],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[1,2,4]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[1, 4] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([1, 2, 4], [1/4, 1/4, 1/2], [1, 4], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], [1, 4], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0028","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93042ee963cd489687313bc2a3b81c2070b08f20a7f559eb9e0e12c26f4a5d1f","family_id":"qf-str22-distribution-event-probability","seed":202610227028,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[2,5],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[2,3,5]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[2, 5] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([2, 3, 5], [1/4, 1/4, 1/2], [2, 5], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], [2, 5], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0029","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bf37d26105dae9527db376e7ce4b35df1763bff13baf8a616e29b973207da2e","family_id":"qf-str22-distribution-event-probability","seed":202610227029,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[3,6],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[3,4,6]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[3, 6] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([3, 4, 6], [1/4, 1/4, 1/2], [3, 6], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], [3, 6], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0030","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6285b733636084d9587e59bba2b40364110768f506c7317f5babd17e0ed86997","family_id":"qf-str22-distribution-event-probability","seed":202610227030,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[4,7],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[4,5,7]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[4, 7] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([4, 5, 7], [1/4, 1/4, 1/2], [4, 7], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], [4, 7], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0031","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50b0fc8cb67a08f0e3a07425f76a9ee38eb9ee4569355e027f2311d93ce59a09","family_id":"qf-str22-distribution-event-probability","seed":202610227031,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[5,8],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[5,6,8]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[5, 8] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([5, 6, 8], [1/4, 1/4, 1/2], [5, 8], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/2], [5, 8], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0032","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6140d892a5457240770a512df17dbcaa8cf18fee31440db2098d82eae2d2b212","family_id":"qf-str22-distribution-event-probability","seed":202610227032,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[6,9],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[6,7,9]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[6, 9] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([6, 7, 9], [1/4, 1/4, 1/2], [6, 9], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], [6, 9], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0033","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9eaabab99494486520a618ebf3bd19ae8a29bec9640daa556b8eaf00fa70f6c8","family_id":"qf-str22-distribution-event-probability","seed":202610227033,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[7,10],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[7,8,10]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[7, 10] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([7, 8, 10], [1/4, 1/4, 1/2], [7, 10], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], [7, 10], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0034","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e37a4457be45af9f4903a67029e9fccfef17835ff40cba5cf7b446fcd5efb19","family_id":"qf-str22-distribution-event-probability","seed":202610227034,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[8,11],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[8,9,11]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[8, 11] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([8, 9, 11], [1/4, 1/4, 1/2], [8, 11], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], [8, 11], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0035","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"58208afae26e8b1ff8db66a5564138bfc92e78ccd34af8a5a39b3025e3d8b9ed","family_id":"qf-str22-distribution-event-probability","seed":202610227035,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[9,12],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[9,10,12]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[9, 12] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([9, 10, 12], [1/4, 1/4, 1/2], [9, 12], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], [9, 12], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0036","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"159da2a6ad229c7e708dc5d6fcd5f793bb6a9f3be7a1e185875442dbfc66f288","family_id":"qf-str22-distribution-event-probability","seed":202610227036,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[10,13],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[10,11,13]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[10, 13] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([10, 11, 13], [1/4, 1/4, 1/2], [10, 13], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/2], [10, 13], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0037","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"20ca6be30fcc1c12655f99f8c98847c76c831b0cea78e7f5472ad302a93a3c2c","family_id":"qf-str22-distribution-event-probability","seed":202610227037,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[11,14],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[11,12,14]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[11, 14] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([11, 12, 14], [1/4, 1/4, 1/2], [11, 14], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], [11, 14], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0038","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8913da72cb72694343efe4963fa98fcdbaed3798d825123dddeed59172c8cf3","family_id":"qf-str22-distribution-event-probability","seed":202610227038,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[12,15],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[12,13,15]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[12, 15] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([12, 13, 15], [1/4, 1/4, 1/2], [12, 15], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], [12, 15], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0039","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa156353ee3342e37e57a915360f0a597d259e186101931a2390a111bd67654a","family_id":"qf-str22-distribution-event-probability","seed":202610227039,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[13,16],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[13,14,16]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[13, 16] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([13, 14, 16], [1/4, 1/4, 1/2], [13, 16], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], [13, 16], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0040","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38121cff165bd30b8cb309375c2ba7f921740839356b7fd384ca57f4ac4274cd","family_id":"qf-str22-distribution-event-probability","seed":202610227040,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[14,17],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[14,15,17]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[14, 17] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([14, 15, 17], [1/4, 1/4, 1/2], [14, 17], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], [14, 17], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0041","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"89426d506ca3900eacd2982b7b5d013c1c12f5aa4f509908ff333c6a04a2e54e","family_id":"qf-str22-distribution-event-probability","seed":202610227041,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[15,18],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[15,16,18]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[15, 18] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([15, 16, 18], [1/4, 1/4, 1/2], [15, 18], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/2], [15, 18], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0042","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0a8035c36b54eb3c2bacab097ebfcae396c5bbfd9db05e14650a9127022c1d8","family_id":"qf-str22-distribution-event-probability","seed":202610227042,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[16,19],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[16,17,19]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[16, 19] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([16, 17, 19], [1/4, 1/4, 1/2], [16, 19], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], [16, 19], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0043","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0bde1ddef91935be7d043fb7bd09077f21c619e97d15aed98082e694de252288","family_id":"qf-str22-distribution-event-probability","seed":202610227043,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[17,20],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[17,18,20]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[17, 20] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([17, 18, 20], [1/4, 1/4, 1/2], [17, 20], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], [17, 20], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0044","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a8ccb34ebc1ae39359d0b44723fcc7a971206af29ca697bc59f5de3ba1d27337","family_id":"qf-str22-distribution-event-probability","seed":202610227044,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[18,21],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[18,19,21]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[18, 21] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([18, 19, 21], [1/4, 1/4, 1/2], [18, 21], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], [18, 21], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0045","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe70bd139ea8970a9e12710fcc4bdbfc4739a908cfc15ea7fe676b8c7b470c94","family_id":"qf-str22-distribution-event-probability","seed":202610227045,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[19,22],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[19,20,22]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[19, 22] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([19, 20, 22], [1/4, 1/4, 1/2], [19, 22], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], [19, 22], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0046","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bde5f0f23cb4e5d9e68d717955368224d8f9506555d2cd063ce1ccb11ce1593f","family_id":"qf-str22-distribution-event-probability","seed":202610227046,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[20,23],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[20,21,23]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[20, 23] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([20, 21, 23], [1/4, 1/4, 1/2], [20, 23], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/2], [20, 23], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0047","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0030bb1d767fe839a5e5672e0d33d8bb19f58d0d63438273f8b4892cc2c6b80c","family_id":"qf-str22-distribution-event-probability","seed":202610227047,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[21,24],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[21,22,24]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[21, 24] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([21, 22, 24], [1/4, 1/4, 1/2], [21, 24], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], [21, 24], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0048","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21d248ecdcf471319bcfa85f3fd06e2efc741fda51001d10edc5b45efc494b81","family_id":"qf-str22-distribution-event-probability","seed":202610227048,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[22,25],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[22,23,25]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[22, 25] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([22, 23, 25], [1/4, 1/4, 1/2], [22, 25], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], [22, 25], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0049","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"177fbe2dac9ce37a17544bd6cbb63e2dfedf0edcb4b2b75a24b4b9e30b4c040b","family_id":"qf-str22-distribution-event-probability","seed":202610227049,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[23,26],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[23,24,26]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[23, 26] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([23, 24, 26], [1/4, 1/4, 1/2], [23, 26], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], [23, 26], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0050","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1db83ace3ee62fbd93b9acac1330bec9db989131f988840234705cdbee79a8c1","family_id":"qf-str22-distribution-event-probability","seed":202610227050,"semantic_template":"qf-str22-distribution-event-probability/{\"event\":[24,27],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[24,25,27]}"},"fields":{"title":"Sum Only the PMF Masses Whose Outcomes Are in the Event","stem":"Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[24, 27] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).","answer":{"kind":"expression","value":"([24, 25, 27], [1/4, 1/4, 1/2], [24, 27], 3/4)","explanation":"Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], [24, 27], 3/4)."},"skills":["Sum Only the PMF Masses Whose Outcomes Are in the Event"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0051","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2fcc3a361fd8bcff05db3d915401de0081153ccb62edbc7dfcc5cac8b5e31b70","family_id":"qf-str22-distribution-expectation","seed":202610227051,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[0,1,3]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([0, 1, 3], [1/4, 1/4, 1/2], 7/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/2], 7/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0052","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"79efca5de1a265467f8213b3c75e758e001fc804412fe1593e39d6faede69843","family_id":"qf-str22-distribution-expectation","seed":202610227052,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[1,2,4]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([1, 2, 4], [1/4, 1/4, 1/2], 11/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], 11/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0053","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4699a9fc69e275e997a52099dfa0272a075ca7e1b7829733e73da6cd1268539","family_id":"qf-str22-distribution-expectation","seed":202610227053,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[2,3,5]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([2, 3, 5], [1/4, 1/4, 1/2], 15/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], 15/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0054","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"be2602fa76a9b292e7fa9440061854bc50656aba28dc75f57b5edbe9bcdd09e1","family_id":"qf-str22-distribution-expectation","seed":202610227054,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[3,4,6]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([3, 4, 6], [1/4, 1/4, 1/2], 19/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], 19/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0055","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"82f12c9c88760659fbc44d561c9efaa0e9859777032590890721d5202e0fa34a","family_id":"qf-str22-distribution-expectation","seed":202610227055,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[4,5,7]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([4, 5, 7], [1/4, 1/4, 1/2], 23/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], 23/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0056","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4dc3f3e55d888f6c8bed12a6f392371c8e66b7a3bffd56a6e51f6686e5ae3c7","family_id":"qf-str22-distribution-expectation","seed":202610227056,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[5,6,8]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([5, 6, 8], [1/4, 1/4, 1/2], 27/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/2], 27/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0057","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c1ed53914308e13fa8b7718968549a4aa5cdd7e4cdf6d4c4c0e6291111e02ace","family_id":"qf-str22-distribution-expectation","seed":202610227057,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[6,7,9]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([6, 7, 9], [1/4, 1/4, 1/2], 31/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], 31/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0058","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9044948824f025d3a5131b637071ce052317449328fc9cfd5edd8abb35228c59","family_id":"qf-str22-distribution-expectation","seed":202610227058,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[7,8,10]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([7, 8, 10], [1/4, 1/4, 1/2], 35/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], 35/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0059","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bfd00c203b81dc9bdab3ec6e676a4cc271c087adeed088d6ab57dd0bb1f4323e","family_id":"qf-str22-distribution-expectation","seed":202610227059,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[8,9,11]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([8, 9, 11], [1/4, 1/4, 1/2], 39/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], 39/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0060","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"eef1ce4440356ed44522a330368ee92ab75bb70fcecb925107a98b50c2dbd815","family_id":"qf-str22-distribution-expectation","seed":202610227060,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[9,10,12]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([9, 10, 12], [1/4, 1/4, 1/2], 43/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], 43/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0061","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3dd9dc68b2cef17fc25107b82b4507b1ad579d7a024e4967013be2eda3bf8a34","family_id":"qf-str22-distribution-expectation","seed":202610227061,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[10,11,13]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([10, 11, 13], [1/4, 1/4, 1/2], 47/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/2], 47/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0062","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"520aec3382657731805d8a974d1f858a84769f0f7d2d3c1f155c9dc2b2038592","family_id":"qf-str22-distribution-expectation","seed":202610227062,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[11,12,14]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([11, 12, 14], [1/4, 1/4, 1/2], 51/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], 51/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0063","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6ca15665346b59740a8771da500ca493ab1a077de3241cb43c6f2c63f52ed337","family_id":"qf-str22-distribution-expectation","seed":202610227063,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[12,13,15]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([12, 13, 15], [1/4, 1/4, 1/2], 55/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], 55/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0064","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e56e9c0e5d2d19a7f64ac710fbf733502771abbee24a6e5ecbbd305475f30e49","family_id":"qf-str22-distribution-expectation","seed":202610227064,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[13,14,16]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([13, 14, 16], [1/4, 1/4, 1/2], 59/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], 59/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0065","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ca983c05092e47ff7f26839023f172c2c1d1160a1832874f3c0a5878e4f74f94","family_id":"qf-str22-distribution-expectation","seed":202610227065,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[14,15,17]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([14, 15, 17], [1/4, 1/4, 1/2], 63/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], 63/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0066","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"76c7ed0760e7e346fce70106385152227a0b5b6e9ee937eba25bae55b771d0b5","family_id":"qf-str22-distribution-expectation","seed":202610227066,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[15,16,18]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([15, 16, 18], [1/4, 1/4, 1/2], 67/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/2], 67/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0067","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"38ab3c722ab282d08d55bece3c6af8a9211c63605741b1c7078f2828cc8e7690","family_id":"qf-str22-distribution-expectation","seed":202610227067,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[16,17,19]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([16, 17, 19], [1/4, 1/4, 1/2], 71/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], 71/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0068","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7d87b87da0ea5bab9c1988fc68f13bd77a51b37f190514c508cb14f4cb0cf4bf","family_id":"qf-str22-distribution-expectation","seed":202610227068,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[17,18,20]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([17, 18, 20], [1/4, 1/4, 1/2], 75/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], 75/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0069","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a72a38f0f659c6007967a30bb420a41d8adc8f8639e217cb1c06269d6e46d0fe","family_id":"qf-str22-distribution-expectation","seed":202610227069,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[18,19,21]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([18, 19, 21], [1/4, 1/4, 1/2], 79/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], 79/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0070","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"07387212755affa48ef975f8b8cf4f427a9d5e32ea00dcd46617ba9c9312ad96","family_id":"qf-str22-distribution-expectation","seed":202610227070,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[19,20,22]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([19, 20, 22], [1/4, 1/4, 1/2], 83/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], 83/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0071","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd7dce39f5350031a90098102a6ae7170907ed1d9daadb64f8adefe9b52a2028","family_id":"qf-str22-distribution-expectation","seed":202610227071,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[20,21,23]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([20, 21, 23], [1/4, 1/4, 1/2], 87/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/2], 87/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0072","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1f737f0a42aa870f635660050dc73f7c9141d7d9e558148ec5397fa9eb1fddad","family_id":"qf-str22-distribution-expectation","seed":202610227072,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[21,22,24]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([21, 22, 24], [1/4, 1/4, 1/2], 91/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], 91/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0073","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f6b01791c1766a8cd576f0f6d167fc586bc25caf3bae7132de1b8a20c8423ecd","family_id":"qf-str22-distribution-expectation","seed":202610227073,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[22,23,25]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([22, 23, 25], [1/4, 1/4, 1/2], 95/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], 95/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0074","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1c2cba90cda676199460797988de28bb3b6a5944cc7597f6d5cdf5cee70746ae","family_id":"qf-str22-distribution-expectation","seed":202610227074,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[23,24,26]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([23, 24, 26], [1/4, 1/4, 1/2], 99/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], 99/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0075","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7d738752770984bc3410c304a055760e495e3aab025fba8eeeff832e7fc5d16","family_id":"qf-str22-distribution-expectation","seed":202610227075,"semantic_template":"qf-str22-distribution-expectation/{\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[24,25,27]}"},"fields":{"title":"An Exact Finite Weighted Mean, Not an Empirical Fit","stem":"Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.","answer":{"kind":"expression","value":"([24, 25, 27], [1/4, 1/4, 1/2], 103/4)","explanation":"Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], 103/4)."},"skills":["An Exact Finite Weighted Mean, Not an Empirical Fit"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0076","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91520c3a78d720ba273e611bf5fd68f653dd14500629cfc889cc37dcbc933892","family_id":"qf-str22-distribution-pushforward","seed":202610227076,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[0,1,3]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([0, 1, 3], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0077","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9ab2c4cf79a84bb560a919555ce920e27fdf2f05f9892d2b9cd8ec1ff1af68a0","family_id":"qf-str22-distribution-pushforward","seed":202610227077,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[1,2,4]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([1, 2, 4], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0078","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"937ffb1919db0ab629fc0c142dd8acd9eb92d9bab277be643adec66f6ad43460","family_id":"qf-str22-distribution-pushforward","seed":202610227078,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[2,3,5]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([2, 3, 5], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0079","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d4542f3c19c20b10caf7b86e7bffe2b84d8f066910c59ff09a6a20a59dee997","family_id":"qf-str22-distribution-pushforward","seed":202610227079,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[3,4,6]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([3, 4, 6], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0080","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5576e0c40c66f2329afd154fbebb8b8ce63fc53d23addf56874bd033b5420a05","family_id":"qf-str22-distribution-pushforward","seed":202610227080,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[4,5,7]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([4, 5, 7], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0081","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5aeb930ed5ab5385b970c052b19b011850c2883741ab26efa7fcf47e0afcdaa8","family_id":"qf-str22-distribution-pushforward","seed":202610227081,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[5,6,8]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([5, 6, 8], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0082","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6d77a93f942dd0a9414a74b5c68452a1bde23368c5f52d2e77aa8d415039d00","family_id":"qf-str22-distribution-pushforward","seed":202610227082,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[6,7,9]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([6, 7, 9], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0083","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"77487cea66e53c32fc95a2951b9d301ab1e2341e65b8cd2c6a2ccf0831d40169","family_id":"qf-str22-distribution-pushforward","seed":202610227083,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[7,8,10]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([7, 8, 10], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a26215be7c5054ddade175f1cb09af9759047bc14d1a3bf4209141998db8564","family_id":"qf-str22-distribution-pushforward","seed":202610227084,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[8,9,11]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0085","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9b8c5b2c6dc72e924142e26ae745dc2ca6345587eee46b1c0ec102128526705d","family_id":"qf-str22-distribution-pushforward","seed":202610227085,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[9,10,12]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([9, 10, 12], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0086","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c2bd27576dcd65e1647a0c69435b35ef48425e8da860508c00d22d39742394ee","family_id":"qf-str22-distribution-pushforward","seed":202610227086,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[10,11,13]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([10, 11, 13], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0087","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6c5a770e5428b2fac54443283138a48d9885d6decddfa4280173b7a5af99b002","family_id":"qf-str22-distribution-pushforward","seed":202610227087,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[11,12,14]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([11, 12, 14], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0088","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed31dcd554f5c3a263a2c1e194410a2cae8b0c85ee20a5d0c49d767f57a39ece","family_id":"qf-str22-distribution-pushforward","seed":202610227088,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[12,13,15]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([12, 13, 15], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0089","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7853950690d4b5e46ef97e4d64a7faa27125bb146a93293373b2bf6accc150db","family_id":"qf-str22-distribution-pushforward","seed":202610227089,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[13,14,16]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([13, 14, 16], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0090","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36d63012ae367517b34487ac1605cb983b1ebcd9c6f6a22fef80d299405fa23b","family_id":"qf-str22-distribution-pushforward","seed":202610227090,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[14,15,17]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([14, 15, 17], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0091","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0202cd6c214345c49f11e6e411cab9ad7b61f2fb86a93ce399d185610c01ec95","family_id":"qf-str22-distribution-pushforward","seed":202610227091,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[15,16,18]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([15, 16, 18], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0092","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a3055fa64ae6fbe658d46cf99492df10a6f7c531ea039b68b3fb37f53df190e3","family_id":"qf-str22-distribution-pushforward","seed":202610227092,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[16,17,19]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([16, 17, 19], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0093","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e11405322a2a18fcbad47f54fe49f9e2b38189251c77e603fd0ff3c90fc4506","family_id":"qf-str22-distribution-pushforward","seed":202610227093,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[17,18,20]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([17, 18, 20], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0094","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a1798204354b545e3e73a31c1b9721f95c139d648d231c6f0031a6a628b4bde","family_id":"qf-str22-distribution-pushforward","seed":202610227094,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[18,19,21]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([18, 19, 21], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0095","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"37f328a5920cfe6a46f54d92f5684240ab1fb71928241e075d8cf69bd8b40932","family_id":"qf-str22-distribution-pushforward","seed":202610227095,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[19,20,22]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([19, 20, 22], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0096","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dca3028d1b706d71a4b4efd520169afc6ea150d350afac9faca0dafeea2e9fac","family_id":"qf-str22-distribution-pushforward","seed":202610227096,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[20,21,23]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([20, 21, 23], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0097","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a571f6d751ccb1d2472389a1d5028b3322ef77840615632535f59601f490e31","family_id":"qf-str22-distribution-pushforward","seed":202610227097,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[21,22,24]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([21, 22, 24], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0098","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24111a11f156f4af1965323a123bbba5bed42434bb3801f3f828709251aab38e","family_id":"qf-str22-distribution-pushforward","seed":202610227098,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[1,2,1],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[22,23,25]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([22, 23, 25], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0099","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8f4399bddb9f227ec1c1516e5caa0e2cc2a70995c754e6d5d930deb0c03d1a92","family_id":"qf-str22-distribution-pushforward","seed":202610227099,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[2,0,2],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[23,24,26]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([23, 24, 26], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0100","batch_id":"batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"479c360048bc05cb42fa518cca59813fdf5cfffc73995e34731ce5683553f4c7","family_id":"qf-str22-distribution-pushforward","seed":202610227100,"semantic_template":"qf-str22-distribution-pushforward/{\"images\":[0,1,0],\"masses\":[[1,4],[1,4],[1,2]],\"outcomes\":[24,25,27]}"},"fields":{"title":"Merge All Masses with a Common Image","stem":"Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.","answer":{"kind":"expression","value":"([24, 25, 27], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])","explanation":"Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])."},"skills":["Merge All Masses with a Common Image"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}