{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0001","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d7071412b5ebf1f4841496bb0d26e268ee98df92afba9fddc9fe999a45bd6d2","family_id":"qf-mod30-cdf-evaluate","seed":202610306001,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-12,-5,1],\"t\":-12}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-12, -5, 1] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-12. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(-12, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-12, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0002","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"aa207ba9fea8c9eeb1f9be0db5db0653e1ec42422b008fb71965e01355f92cad","family_id":"qf-mod30-cdf-evaluate","seed":202610306002,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-11,-4,2],\"t\":-4}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-11, -4, 2] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-4. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(-4, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-4, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0003","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"16ed5d743deef39a97db5bc039fb25531fd55f5afb8b44663ba8379d2cd22707","family_id":"qf-mod30-cdf-evaluate","seed":202610306003,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-10,-3,3],\"t\":3}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-10, -3, 3] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=3. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(3, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (3, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0004","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cba8fb0f84de064e6505a9e7d7f5025e0f159c9bbf92dc2f705256141963914","family_id":"qf-mod30-cdf-evaluate","seed":202610306004,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-9,-2,4],\"t\":-9}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-9, -2, 4] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-9. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(-9, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-9, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0005","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae14426274e11cdbdaa379bafd7e049c703f403cd8a7e22486100fde0f968be5","family_id":"qf-mod30-cdf-evaluate","seed":202610306005,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-8,-1,5],\"t\":-1}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-8, -1, 5] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-1. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(-1, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-1, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0006","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"afc15c69261903d2d20afccb8acdfea11e84f43fd5da9548e517befe4a4eff20","family_id":"qf-mod30-cdf-evaluate","seed":202610306006,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-7,0,6],\"t\":6}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-7, 0, 6] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=6. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(6, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (6, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0007","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4d8b41d95b033dfcd81f4b90bde70c7c1ec206196ef91e932cb817e3d537561","family_id":"qf-mod30-cdf-evaluate","seed":202610306007,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-6,1,7],\"t\":-6}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-6, 1, 7] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-6. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(-6, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-6, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0008","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7f170cc1e69d9b1045791168d0c400f985c72c3c76fd398e0bb860d87faf2d9","family_id":"qf-mod30-cdf-evaluate","seed":202610306008,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-5,2,8],\"t\":2}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-5, 2, 8] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=2. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(2, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (2, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0009","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa73c74d7b4b396eaf79315a167a7e553fd801d27de67ee896169b2fd37857a9","family_id":"qf-mod30-cdf-evaluate","seed":202610306009,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-4,3,9],\"t\":9}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-4, 3, 9] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(9, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (9, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0010","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e6ac7f60995073980b6dbccffa0352d8b3a7cbd83d37f82adac00ac84c6d6638","family_id":"qf-mod30-cdf-evaluate","seed":202610306010,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-3,4,10],\"t\":-3}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-3, 4, 10] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-3. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(-3, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (-3, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0011","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af6368f2aae9ff5e234e04c4ca23631028b2b7c2af38551a8134166ff8c2d3ad","family_id":"qf-mod30-cdf-evaluate","seed":202610306011,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-2,5,11],\"t\":5}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-2, 5, 11] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=5. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(5, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (5, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0012","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b7ff2b6d4c46b18f5f0a7cec6be771af1c4120530a477f9e28df2fd912804e0","family_id":"qf-mod30-cdf-evaluate","seed":202610306012,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-1,6,12],\"t\":12}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[-1, 6, 12] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=12. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(12, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (12, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0013","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a3accdb8e93b2166166c1f7a56667f95374c7bddd3bbc946ee3c7fecb72876e","family_id":"qf-mod30-cdf-evaluate","seed":202610306013,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[0,7,13],\"t\":0}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[0, 7, 13] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=0. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(0, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (0, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0014","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9894550e96a338fc220a892c2debfd1478fa3adba987deb4752a6954b519f76b","family_id":"qf-mod30-cdf-evaluate","seed":202610306014,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[1,8,14],\"t\":8}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[1, 8, 14] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=8. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(8, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (8, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0015","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4433e22d2430cbe4f84d46773956e32e1959de107cda2100aa8d97c6122a7d5c","family_id":"qf-mod30-cdf-evaluate","seed":202610306015,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[2,9,15],\"t\":15}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[2, 9, 15] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=15. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(15, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (15, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0016","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ad782b1a657b3ecb037c14f924ec5024b1ab5ac4c4c462550537a19c8abca51d","family_id":"qf-mod30-cdf-evaluate","seed":202610306016,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[3,10,16],\"t\":3}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[3, 10, 16] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=3. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(3, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (3, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0017","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5e4ac3738b310c03b5d350b57b3f6ef000cd383add7163c349ef0ffd7b77d8c0","family_id":"qf-mod30-cdf-evaluate","seed":202610306017,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[4,11,17],\"t\":11}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[4, 11, 17] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=11. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(11, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (11, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0018","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"24a89187541185455a4e749a0bff0a716804f953d598c82beef6df6eae3ca374","family_id":"qf-mod30-cdf-evaluate","seed":202610306018,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[5,12,18],\"t\":18}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[5, 12, 18] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=18. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(18, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (18, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0019","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cdfacec1d3fa61cc921fde6b0c1cc221d8f60e6efd09052234fdebaa6ca8ac2","family_id":"qf-mod30-cdf-evaluate","seed":202610306019,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[6,13,19],\"t\":6}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[6, 13, 19] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=6. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(6, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (6, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0020","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21acbcdc840a042de46d6c38a10f9d372f0dface32405def6ad9fd641fc826d2","family_id":"qf-mod30-cdf-evaluate","seed":202610306020,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[7,14,20],\"t\":14}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[7, 14, 20] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=14. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(14, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (14, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0021","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d3b85f279285b76186dd0ef79357958b2b1c5e3e579ce67d86833f5548fd760d","family_id":"qf-mod30-cdf-evaluate","seed":202610306021,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[8,15,21],\"t\":21}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[8, 15, 21] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=21. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(21, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (21, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0022","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"edf7c9a45c4aee01788af274d9be2f5755f049ac6dedc7413f62d4de81fb115e","family_id":"qf-mod30-cdf-evaluate","seed":202610306022,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[9,16,22],\"t\":9}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[9, 16, 22] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(9, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (9, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0023","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a79a846c0676ca72212cf45dd0c83feb0988f0ce47c82a0d7d964fddb3514913","family_id":"qf-mod30-cdf-evaluate","seed":202610306023,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[10,17,23],\"t\":17}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[10, 17, 23] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=17. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(17, 1/2, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (17, 1/2, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0024","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"255d111697cd7441852366803934b1d843b5da22a19c47e2d6a5c932d10c6414","family_id":"qf-mod30-cdf-evaluate","seed":202610306024,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[11,18,24],\"t\":24}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[11, 18, 24] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=24. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(24, 1, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (24, 1, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0025","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"901a4b51bbf382ea9de3759dd145d427ee77ffca120de559fb01819b73a3b7e6","family_id":"qf-mod30-cdf-evaluate","seed":202610306025,"semantic_template":"qf-mod30-cdf-evaluate/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[12,19,25],\"t\":12}"},"fields":{"title":"The CDF Includes Mass Exactly at the Threshold","stem":"Use the complete finite discrete law with distinct integer support=[12, 19, 25] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=12. Give (t,F_t,uses_X_less_than_or_equal_t).","answer":{"kind":"expression","value":"(12, 1/4, uses_X_less_than_or_equal_t)","explanation":"Sum every mass whose aligned support value is at most t, including equality. Below all support the sum is zero; at or above the largest support it is one. The exact result, retaining the requested order, is (12, 1/4, uses_X_less_than_or_equal_t)."},"skills":["The CDF Includes Mass Exactly at the Threshold"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0026","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5857264f7973f3057b6e9ced251ddc8ba688ea82a0e4500e610be7243e79714","family_id":"qf-mod30-cdf-jump","seed":202610306026,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-12,-5,1],\"t\":-12}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-12, -5, 1] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-12. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(-12, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-12, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0027","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4b233e4714edf9c6e87be77da1ae53a9e6a2c69319455e7262327f630c641c12","family_id":"qf-mod30-cdf-jump","seed":202610306027,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-11,-4,2],\"t\":-4}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-11, -4, 2] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-4. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(-4, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-4, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0028","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e830f88640a0627a386e6bc6d1b3154b91969d16cb61d910bec9e3380afb0751","family_id":"qf-mod30-cdf-jump","seed":202610306028,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-10,-3,3],\"t\":3}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-10, -3, 3] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=3. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(3, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (3, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0029","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ee63e11d93c93e2a51f3c3d31168f70948ba58e404382644d632382d33703f8","family_id":"qf-mod30-cdf-jump","seed":202610306029,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-9,-2,4],\"t\":-9}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-9, -2, 4] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-9. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(-9, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-9, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0030","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec51c3de577c8f8e967480c9bbb0f0282ada782fcf40b32ba21260899a08d354","family_id":"qf-mod30-cdf-jump","seed":202610306030,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-8,-1,5],\"t\":-1}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-8, -1, 5] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-1. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(-1, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-1, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0031","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b02691c1688f97246c97de17d4dfed1733ca950ccbc6392026c965b47454506","family_id":"qf-mod30-cdf-jump","seed":202610306031,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-7,0,6],\"t\":6}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-7, 0, 6] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=6. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(6, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (6, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0032","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ae6a6d95e72797b3e5d10ffcf1a9fdff797ecab56b7e730298e9be5b15aab3e6","family_id":"qf-mod30-cdf-jump","seed":202610306032,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-6,1,7],\"t\":-6}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-6, 1, 7] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-6. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(-6, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-6, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0033","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"33a19cdd05e2d4450032742f71accbacb77dea8c79c7082ed868be0dabb34e76","family_id":"qf-mod30-cdf-jump","seed":202610306033,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-5,2,8],\"t\":2}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-5, 2, 8] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=2. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(2, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (2, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0034","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f4c6466813850a91050a873a879fd447f3913d1cfb87aaed81975caaaf5a207","family_id":"qf-mod30-cdf-jump","seed":202610306034,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-4,3,9],\"t\":9}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-4, 3, 9] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(9, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (9, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0035","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e74e483525554ed13eafec2f753acccdad64fc115e480cc8b187fff6e63b8c8c","family_id":"qf-mod30-cdf-jump","seed":202610306035,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-3,4,10],\"t\":-3}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-3, 4, 10] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-3. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(-3, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (-3, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0036","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a5f97f883e964f3f7d3546133e7b765708ec001d1eb559599158f61fc23f1da","family_id":"qf-mod30-cdf-jump","seed":202610306036,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-2,5,11],\"t\":5}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-2, 5, 11] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=5. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(5, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (5, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0037","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7af4b33c9b4820e690eeb8ee04d77f27c927930a17f40eefa1a290025e5fa328","family_id":"qf-mod30-cdf-jump","seed":202610306037,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-1,6,12],\"t\":12}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[-1, 6, 12] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=12. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(12, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (12, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0038","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"938755bc6c93c1842614946a67bd855698feade1d565f4bd6fa8cd104f6e03ea","family_id":"qf-mod30-cdf-jump","seed":202610306038,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[0,7,13],\"t\":0}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[0, 7, 13] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=0. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(0, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (0, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0039","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5abeb8f1fe489e9a0ba3f0cd641fb7e9022cf20e889760719f198a9224d84db","family_id":"qf-mod30-cdf-jump","seed":202610306039,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[1,8,14],\"t\":8}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[1, 8, 14] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=8. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(8, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (8, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0040","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"36b97f1515c3a14b20926c9fb4f1614e6d6f4cda5818731c56f25014d5428c65","family_id":"qf-mod30-cdf-jump","seed":202610306040,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[2,9,15],\"t\":15}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[2, 9, 15] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=15. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(15, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (15, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0041","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"63664dfd13ca33d09cc39ceb387eccca9af89b06f5b29071d316fbb8ec222db0","family_id":"qf-mod30-cdf-jump","seed":202610306041,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[3,10,16],\"t\":3}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[3, 10, 16] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=3. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(3, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (3, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0042","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ba262efc5a1310e944ba21c6cfafacddee5f6c584146fbfc1b62f52f37f49c9","family_id":"qf-mod30-cdf-jump","seed":202610306042,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[4,11,17],\"t\":11}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[4, 11, 17] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=11. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(11, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (11, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0043","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f68f6b569d2f546ce6c99c5720cd8b4f1c58686755fb2ada322008a6f4a68ecf","family_id":"qf-mod30-cdf-jump","seed":202610306043,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[5,12,18],\"t\":18}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[5, 12, 18] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=18. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(18, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (18, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0044","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c9501a3667e767582a356f84983931154ac620e14e0d10170bf24848e306a79f","family_id":"qf-mod30-cdf-jump","seed":202610306044,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[6,13,19],\"t\":6}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[6, 13, 19] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=6. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(6, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (6, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0045","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"995044fbc59f619512d28ba0deb09f25dda1fe07af6239fddeff4fd42c8b59ba","family_id":"qf-mod30-cdf-jump","seed":202610306045,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[7,14,20],\"t\":14}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[7, 14, 20] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=14. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(14, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (14, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0046","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"73fa5412dd51f1297830aaa0a7eaefcd7f4586617f8cc10de96e8d412e5922d0","family_id":"qf-mod30-cdf-jump","seed":202610306046,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[8,15,21],\"t\":21}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[8, 15, 21] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=21. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(21, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (21, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0047","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a522e6332e07ca6f8837048ee3e139d2b8fcb8c3bf7e37c7b3586411fb977af5","family_id":"qf-mod30-cdf-jump","seed":202610306047,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[9,16,22],\"t\":9}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[9, 16, 22] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(9, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (9, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0048","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f45f9c414661f9ea39262f13a7cf12c64ae9f4931c34c44e3ae7903fe79c6fff","family_id":"qf-mod30-cdf-jump","seed":202610306048,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[10,17,23],\"t\":17}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[10, 17, 23] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=17. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(17, 1/4, 1/2, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (17, 1/4, 1/2, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0049","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d194c65fd028e9a00482f55db89f8a5c11d7f645eb5cceaacb08338fa299a039","family_id":"qf-mod30-cdf-jump","seed":202610306049,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[11,18,24],\"t\":24}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[11, 18, 24] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=24. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(24, 1/2, 1, 1/2, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (24, 1/2, 1, 1/2, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0050","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c5a71e56bfce7e14e8e8bacea4c3458f8a867c05cb2ea00f16919e6484226778","family_id":"qf-mod30-cdf-jump","seed":202610306050,"semantic_template":"qf-mod30-cdf-jump/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[12,19,25],\"t\":12}"},"fields":{"title":"The CDF Jump Is the Point Mass, Not the Whole CDF Value","stem":"Use the complete finite discrete law with distinct integer support=[12, 19, 25] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=12. Give (t,left_limit,F_t,point_mass,jump_identity?).","answer":{"kind":"expression","value":"(12, 0, 1/4, 1/4, Yes)","explanation":"Sum support strictly below t for the left limit, include equality for F(t), and use zero point mass when t is absent. Compare left_limit+point_mass with F(t) exactly. The exact result, retaining the requested order, is (12, 0, 1/4, 1/4, Yes)."},"skills":["The CDF Jump Is the Point Mass, Not the Whole CDF Value"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0051","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8cedde6fdc40d14f0946703ca1d76e2c4105d2015cf92833da4b573babecd3a3","family_id":"qf-mod30-cdf-monotonicity","seed":202610306051,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-12,\"masses\":[[1,4],[1,4],[1,2]],\"right\":1,\"support\":[-12,-5,1]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-12, -5, 1] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-12≤right=1. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-12, 1, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-12, 1, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0052","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"249a5d5454613edc5d7c5449926e843ccad5a7d044b8f8295ba1e727237f2959","family_id":"qf-mod30-cdf-monotonicity","seed":202610306052,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-11,\"masses\":[[1,4],[1,4],[1,2]],\"right\":2,\"support\":[-11,-4,2]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-11, -4, 2] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-11≤right=2. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-11, 2, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-11, 2, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0053","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74f998192d8809e5728d119f673a5dcbf328658ff5fdd4e123600d5d685d41d5","family_id":"qf-mod30-cdf-monotonicity","seed":202610306053,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-10,\"masses\":[[1,4],[1,4],[1,2]],\"right\":3,\"support\":[-10,-3,3]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-10, -3, 3] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-10≤right=3. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-10, 3, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-10, 3, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0054","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb692e3a718933e89a201a9208a7b8afc4e57783036c7ce4bf5b22dc2251b428","family_id":"qf-mod30-cdf-monotonicity","seed":202610306054,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-9,\"masses\":[[1,4],[1,4],[1,2]],\"right\":4,\"support\":[-9,-2,4]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-9, -2, 4] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-9≤right=4. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-9, 4, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-9, 4, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0055","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2700a20f959246a5e189a87c49d4bf4fc45aa9f754c3bf41ef559851f0ab089e","family_id":"qf-mod30-cdf-monotonicity","seed":202610306055,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-8,\"masses\":[[1,4],[1,4],[1,2]],\"right\":5,\"support\":[-8,-1,5]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-8, -1, 5] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-8≤right=5. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-8, 5, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-8, 5, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0056","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2255ec477c275fdd395ec8419d8126406791112acfdc3a498b9b94ffbf90543d","family_id":"qf-mod30-cdf-monotonicity","seed":202610306056,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-7,\"masses\":[[1,4],[1,4],[1,2]],\"right\":6,\"support\":[-7,0,6]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-7, 0, 6] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-7≤right=6. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-7, 6, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-7, 6, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0057","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d6521470894ea1fc2091adb4c862fc6f5d7f67d4d86d806b49076c738033a548","family_id":"qf-mod30-cdf-monotonicity","seed":202610306057,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-6,\"masses\":[[1,4],[1,4],[1,2]],\"right\":7,\"support\":[-6,1,7]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-6, 1, 7] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-6≤right=7. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-6, 7, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-6, 7, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0058","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2df2d458db17fe5bd080c54990effa3072938a7b387137f84fc7e98230b70124","family_id":"qf-mod30-cdf-monotonicity","seed":202610306058,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-5,\"masses\":[[1,4],[1,4],[1,2]],\"right\":8,\"support\":[-5,2,8]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-5, 2, 8] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-5≤right=8. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-5, 8, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-5, 8, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0059","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"acdf66991260d7c3f0c4304037a192227adde3246f7e4c2e04710b53e3586775","family_id":"qf-mod30-cdf-monotonicity","seed":202610306059,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-4,\"masses\":[[1,4],[1,4],[1,2]],\"right\":9,\"support\":[-4,3,9]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-4, 3, 9] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-4≤right=9. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-4, 9, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-4, 9, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0060","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91016bf91266974e3368c986949a28aefe1eeb0574e6f3dd6983defb148b4b25","family_id":"qf-mod30-cdf-monotonicity","seed":202610306060,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-3,\"masses\":[[1,4],[1,4],[1,2]],\"right\":10,\"support\":[-3,4,10]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-3, 4, 10] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-3≤right=10. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-3, 10, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-3, 10, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0061","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"402c03a841dc1e90c32484ef369874fef18eec41d3b6e738e9967b1813a55ff2","family_id":"qf-mod30-cdf-monotonicity","seed":202610306061,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-2,\"masses\":[[1,4],[1,4],[1,2]],\"right\":11,\"support\":[-2,5,11]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-2, 5, 11] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-2≤right=11. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-2, 11, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-2, 11, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0062","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ac5761168d61fcfce4dc3915c719b90e19ce1327287f8842ca5f8e3d717c1368","family_id":"qf-mod30-cdf-monotonicity","seed":202610306062,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":-1,\"masses\":[[1,4],[1,4],[1,2]],\"right\":12,\"support\":[-1,6,12]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[-1, 6, 12] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=-1≤right=12. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(-1, 12, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (-1, 12, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0063","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"74b08f5716b1f790ba2992201c7c89d79eff7861a318f1febf1068ce652bdaff","family_id":"qf-mod30-cdf-monotonicity","seed":202610306063,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":0,\"masses\":[[1,4],[1,4],[1,2]],\"right\":13,\"support\":[0,7,13]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[0, 7, 13] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=0≤right=13. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(0, 13, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (0, 13, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0064","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ec696c87407a8307ae03924b3691591be2ffd52720c3897870269414a1431088","family_id":"qf-mod30-cdf-monotonicity","seed":202610306064,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":1,\"masses\":[[1,4],[1,4],[1,2]],\"right\":14,\"support\":[1,8,14]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[1, 8, 14] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=1≤right=14. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(1, 14, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (1, 14, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0065","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8219ff1d474684c7b16d8b2a15b4d3faf3557c41e89a3a55e8c8f68dcd1c32a6","family_id":"qf-mod30-cdf-monotonicity","seed":202610306065,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":2,\"masses\":[[1,4],[1,4],[1,2]],\"right\":15,\"support\":[2,9,15]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[2, 9, 15] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=2≤right=15. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(2, 15, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (2, 15, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0066","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee4f0dd3b796139dcff3e6e1eff43c8dca42baea815df9abd7304f95f1d88d60","family_id":"qf-mod30-cdf-monotonicity","seed":202610306066,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":3,\"masses\":[[1,4],[1,4],[1,2]],\"right\":16,\"support\":[3,10,16]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[3, 10, 16] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=3≤right=16. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(3, 16, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (3, 16, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0067","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b7baa5f15b93adba4fd634f57293855fcde4b6f30688d17fa5a33082c55970c9","family_id":"qf-mod30-cdf-monotonicity","seed":202610306067,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":4,\"masses\":[[1,4],[1,4],[1,2]],\"right\":17,\"support\":[4,11,17]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[4, 11, 17] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=4≤right=17. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(4, 17, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (4, 17, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0068","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9edd45097f3e31de7f1f456c9f4e5e18642f8fc6e913cb7aef18a99e87759cc0","family_id":"qf-mod30-cdf-monotonicity","seed":202610306068,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":5,\"masses\":[[1,4],[1,4],[1,2]],\"right\":18,\"support\":[5,12,18]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[5, 12, 18] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=5≤right=18. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(5, 18, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (5, 18, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0069","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"71ffbd8b3ceffd18e84f6dff039f65e9e8be6aab844bd2f00c68463a86d66c1d","family_id":"qf-mod30-cdf-monotonicity","seed":202610306069,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":6,\"masses\":[[1,4],[1,4],[1,2]],\"right\":19,\"support\":[6,13,19]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[6, 13, 19] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=6≤right=19. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(6, 19, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (6, 19, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0070","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cd19579a9146ac039c37b22206b4cc879be0018e6b9acdc2b93730043515d598","family_id":"qf-mod30-cdf-monotonicity","seed":202610306070,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":7,\"masses\":[[1,4],[1,4],[1,2]],\"right\":20,\"support\":[7,14,20]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[7, 14, 20] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=7≤right=20. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(7, 20, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (7, 20, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0071","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0bd2268f262f4dd84c149d49492ae023f3cfe5ba1e3297fcdfd7953a30d2d8d","family_id":"qf-mod30-cdf-monotonicity","seed":202610306071,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":8,\"masses\":[[1,4],[1,4],[1,2]],\"right\":21,\"support\":[8,15,21]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[8, 15, 21] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=8≤right=21. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(8, 21, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (8, 21, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0072","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df3906465d05f3c83d396a5215cd5c24ed28cf51a6730e8aec3e1ca8fa1434e6","family_id":"qf-mod30-cdf-monotonicity","seed":202610306072,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":9,\"masses\":[[1,4],[1,4],[1,2]],\"right\":22,\"support\":[9,16,22]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[9, 16, 22] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=9≤right=22. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(9, 22, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (9, 22, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0073","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6366d149ecf9bdcf9d411ece3312340cbdae76fea3d6df808372f903a920c437","family_id":"qf-mod30-cdf-monotonicity","seed":202610306073,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":10,\"masses\":[[1,4],[1,4],[1,2]],\"right\":23,\"support\":[10,17,23]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[10, 17, 23] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=10≤right=23. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(10, 23, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (10, 23, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0074","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c90d7c7de582e5be18eeb57ca2a389a1bb04c870a4456bd2847c36cd8080250f","family_id":"qf-mod30-cdf-monotonicity","seed":202610306074,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":11,\"masses\":[[1,4],[1,4],[1,2]],\"right\":24,\"support\":[11,18,24]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[11, 18, 24] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=11≤right=24. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(11, 24, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (11, 24, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0075","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fdc1330ccbbe68267eba6485ed95f5dc7bfd4bac02aa098a3cc82a7a00647287","family_id":"qf-mod30-cdf-monotonicity","seed":202610306075,"semantic_template":"qf-mod30-cdf-monotonicity/{\"left\":12,\"masses\":[[1,4],[1,4],[1,2]],\"right\":25,\"support\":[12,19,25]}"},"fields":{"title":"Nested Threshold Events Give a Nondecreasing CDF","stem":"Use the complete finite discrete law with distinct integer support=[12, 19, 25] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Ordered thresholds left=12≤right=25. Give (left,right,F_left,F_right,monotone?).","answer":{"kind":"expression","value":"(12, 25, 1/4, 1, Yes)","explanation":"Aggregate both cumulative masses from the same complete law and compare exact rationals. Reversing the threshold premise is rejected; sampled values alone are not a general CDF construction. The exact result, retaining the requested order, is (12, 25, 1/4, 1, Yes)."},"skills":["Nested Threshold Events Give a Nondecreasing CDF"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0076","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ed032694461d28e2c8e68d68a07ea3fc56c4b5490da426e5204781259d2f514d","family_id":"qf-mod30-cdf-right-continuity","seed":202610306076,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-12,-5,1],\"t\":-12}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-12, -5, 1] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-12. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(-12, 1/4, -5, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-12, 1/4, -5, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0077","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3a724120f13d3246fc893e04ce3adc441cda6cfe791da46f1ff87c327101c47a","family_id":"qf-mod30-cdf-right-continuity","seed":202610306077,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-11,-4,2],\"t\":-4}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-11, -4, 2] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-4. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(-4, 1/2, 2, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-4, 1/2, 2, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0078","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"55c0d821a45bd21cce9b8a1ec235a2adf0761c4c8e92e78bc933412cd1570907","family_id":"qf-mod30-cdf-right-continuity","seed":202610306078,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-10,-3,3],\"t\":3}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-10, -3, 3] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=3. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(3, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (3, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0079","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b58237b872a3b2dc6e94e9166205c8480466d6d0253dfe17df6449b753dae7f5","family_id":"qf-mod30-cdf-right-continuity","seed":202610306079,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-9,-2,4],\"t\":-9}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-9, -2, 4] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-9. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(-9, 1/4, -2, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-9, 1/4, -2, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0080","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7218061036fa8549fa3589bcf7f4e04b3f4349fddbb7e4c7ed56fc71832e13b6","family_id":"qf-mod30-cdf-right-continuity","seed":202610306080,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-8,-1,5],\"t\":-1}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-8, -1, 5] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-1. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(-1, 1/2, 5, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-1, 1/2, 5, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0081","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3aeb7b71a37bd72817d2d89c161824b7398dafe477e5d8be2441084af878a97e","family_id":"qf-mod30-cdf-right-continuity","seed":202610306081,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-7,0,6],\"t\":6}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-7, 0, 6] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=6. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(6, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (6, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0082","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e6174ae711c7b49887849dae40842f80ba9fcf2ff7914ead9ca0ebbe037819","family_id":"qf-mod30-cdf-right-continuity","seed":202610306082,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-6,1,7],\"t\":-6}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-6, 1, 7] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-6. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(-6, 1/4, 1, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-6, 1/4, 1, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0083","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2aab1e0886f817897e55981caba623cab77f3879f2d0a1799720220cdc5f09c6","family_id":"qf-mod30-cdf-right-continuity","seed":202610306083,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-5,2,8],\"t\":2}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-5, 2, 8] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=2. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(2, 1/2, 8, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (2, 1/2, 8, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0084","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"053deb837222fea37db62d6a60dc0155824a9dd84fa904673ebee589c03b0f9c","family_id":"qf-mod30-cdf-right-continuity","seed":202610306084,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-4,3,9],\"t\":9}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-4, 3, 9] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(9, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (9, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0085","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8a5f2e03523595fec9cdae41b7562a40c9fa630906eaf3d964d30ef1fad942a5","family_id":"qf-mod30-cdf-right-continuity","seed":202610306085,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-3,4,10],\"t\":-3}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-3, 4, 10] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=-3. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(-3, 1/4, 4, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (-3, 1/4, 4, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0086","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e291df3a0a15f7a44d177610c6957ae18484e3a8b9feb2b64b527ad7dbe300b9","family_id":"qf-mod30-cdf-right-continuity","seed":202610306086,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-2,5,11],\"t\":5}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-2, 5, 11] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=5. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(5, 1/2, 11, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (5, 1/2, 11, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0087","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b91ee1374727ab0b8243252d485518e6511ae7e4cde6eeea7551418fa1fe45cc","family_id":"qf-mod30-cdf-right-continuity","seed":202610306087,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[-1,6,12],\"t\":12}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[-1, 6, 12] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=12. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(12, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (12, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0088","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d13ceb8345a4f312851fdee94bc0e1ea96e262749ad2b603c339e0cd4d9e99d5","family_id":"qf-mod30-cdf-right-continuity","seed":202610306088,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[0,7,13],\"t\":0}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[0, 7, 13] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=0. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(0, 1/4, 7, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (0, 1/4, 7, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0089","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d2553dafbcf6526a5af44e2509e54e4b997ad04a4b47a5e3009391ba47083949","family_id":"qf-mod30-cdf-right-continuity","seed":202610306089,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[1,8,14],\"t\":8}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[1, 8, 14] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=8. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(8, 1/2, 14, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (8, 1/2, 14, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0090","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"611259e1a2e9b7fd0a684e9d406581ad41c9bf610db769c8d77af253c8e3e60c","family_id":"qf-mod30-cdf-right-continuity","seed":202610306090,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[2,9,15],\"t\":15}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[2, 9, 15] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=15. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(15, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (15, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0091","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c07675235ef1e742743cebc562b1a2ca48464842e887748f408dcf60dbdbf24a","family_id":"qf-mod30-cdf-right-continuity","seed":202610306091,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[3,10,16],\"t\":3}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[3, 10, 16] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=3. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(3, 1/4, 10, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (3, 1/4, 10, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0092","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"612908f9fcdda6530f98a59872dd6a88239bf104e89584389f6889e1105b63b1","family_id":"qf-mod30-cdf-right-continuity","seed":202610306092,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[4,11,17],\"t\":11}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[4, 11, 17] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=11. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(11, 1/2, 17, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (11, 1/2, 17, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0093","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cca607b4537ebb6f99f0004e43e6a5e19365300d4918fabb472eb2b45ec66283","family_id":"qf-mod30-cdf-right-continuity","seed":202610306093,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[5,12,18],\"t\":18}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[5, 12, 18] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=18. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(18, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (18, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0094","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"587259d054c76344b1fa4d33ec4c4a8f2c47f7e59c21084d59c6066e3dd63897","family_id":"qf-mod30-cdf-right-continuity","seed":202610306094,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[6,13,19],\"t\":6}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[6, 13, 19] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=6. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(6, 1/4, 13, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (6, 1/4, 13, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0095","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ac7dd9e465719994ced92683ab8df6bf316dc13fcc7063edb845a5877597527","family_id":"qf-mod30-cdf-right-continuity","seed":202610306095,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[7,14,20],\"t\":14}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[7, 14, 20] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=14. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(14, 1/2, 20, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (14, 1/2, 20, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0096","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c83db9e01656e2cf3416e287f4a8b0f7555070e353edcc2d20315dfc56d88371","family_id":"qf-mod30-cdf-right-continuity","seed":202610306096,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[8,15,21],\"t\":21}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[8, 15, 21] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=21. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(21, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (21, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0097","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2e733b92b8801a12341191c0cec6c92665712095fe85063b3dade97831293dbd","family_id":"qf-mod30-cdf-right-continuity","seed":202610306097,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[9,16,22],\"t\":9}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[9, 16, 22] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=9. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(9, 1/4, 16, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (9, 1/4, 16, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0098","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"101984ef6c69b499605db1ef87b6ac028dd9f503b24f0ddc9c2cf1e1b562cc49","family_id":"qf-mod30-cdf-right-continuity","seed":202610306098,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[10,17,23],\"t\":17}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[10, 17, 23] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=17. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(17, 1/2, 23, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (17, 1/2, 23, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0099","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"864a0d85744a911946b0a1be4d60f46f9350a4ad1f0e3250feac67d2a3a524d1","family_id":"qf-mod30-cdf-right-continuity","seed":202610306099,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[11,18,24],\"t\":24}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[11, 18, 24] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=24. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(24, 1, null, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (24, 1, null, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261030-r06-cumulative-distribution-function-0100","batch_id":"batch-20261014-advance-20261030-r06-cumulative-distribution-function-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f4772212ac3e989f8d2a31865a8a06dc8abd0f0cd8bba147bd36efce887de25f","family_id":"qf-mod30-cdf-right-continuity","seed":202610306100,"semantic_template":"qf-mod30-cdf-right-continuity/{\"masses\":[[1,4],[1,4],[1,2]],\"support\":[12,19,25],\"t\":12}"},"fields":{"title":"A Finite-Law CDF Stays Constant Just to the Right","stem":"Use the complete finite discrete law with distinct integer support=[12, 19, 25] and aligned nonnegative exact masses=[1/4, 1/4, 1/2], summing exactly to 1. Preserve support-to-mass alignment even when support is not sorted. F(t)=P(X≤t) includes equality. The CDF is not a point mass or density: its jump at t is P(X=t), it is nondecreasing and right-continuous, with endpoint limits 0 and 1. Keep exact reduced rational outputs with positive denominators, omitting /1 for integers. Threshold t=12. Give (t,F_t,next_support,constant_on_right_neighborhood_before_next_support,right_continuous?). next_support is the least support value strictly above t, or null when none exists.","answer":{"kind":"expression","value":"(12, 1/4, 19, constant_on_right_neighborhood_before_next_support, Yes)","explanation":"The right value includes any mass at t and remains unchanged before the next support point. With no next support it remains constant for all larger thresholds. This is right continuity, not left continuity or a density claim. The exact result, retaining the requested order, is (12, 1/4, 19, constant_on_right_neighborhood_before_next_support, Yes)."},"skills":["A Finite-Law CDF Stays Constant Just to the Right"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}