{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261014-advance-20261106-r04-estimator-variance-0001","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcb78e820fc4752667a7e079cfb6f60faf9c3d9c6bffe688a5fb07209a93c79d","family_id":"qf-mod37-estvar-at-parameter","seed":202611064001,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"parameter\":0,\"statistic\":[0,1,2,3]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 2, 1, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 2, 1, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0002","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"21cbd63ba8bfd22d5172e62330cd9b367cbdd4f077665af0c91b199c098c1d5b","family_id":"qf-mod37-estvar-at-parameter","seed":202611064002,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"parameter\":1,\"statistic\":[0,1,3,4]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 13/5, 56/25, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 13/5, 56/25, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0003","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f799b530269850861088c4c673d71870e8cee722b9ab51ecd6e8146791d1ee76","family_id":"qf-mod37-estvar-at-parameter","seed":202611064003,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"parameter\":0,\"statistic\":[0,1,4,5]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 11/3, 29/9, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 11/3, 29/9, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0004","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"62cf761f925d51889146ab7b3edf1a82261ad45e5e608d1be53b2319cc096ff4","family_id":"qf-mod37-estvar-at-parameter","seed":202611064004,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"parameter\":1,\"statistic\":[0,1,5,6]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 71/17, 1606/289, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 71/17, 1606/289, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0005","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"df64ed0776816cd557e93776e09419096d053378f795cdb46c69b106f0e1a6f5","family_id":"qf-mod37-estvar-at-parameter","seed":202611064005,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"parameter\":0,\"statistic\":[0,1,6,7]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 38/7, 313/49, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 38/7, 313/49, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0006","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"78306a6b99bbb494ca84410089bd408eadd2756491c9956601ff6ba92b7640ea","family_id":"qf-mod37-estvar-at-parameter","seed":202611064006,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"parameter\":1,\"statistic\":[0,1,7,8]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 111/19, 3620/361, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 111/19, 3620/361, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0007","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b701da4474407943f6f70abfd45e46598e8c65cca3f9a474ba2e22a8f26a86a0","family_id":"qf-mod37-estvar-at-parameter","seed":202611064007,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"parameter\":0,\"statistic\":[0,1,8,9]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 29/4, 163/16, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 29/4, 163/16, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0008","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6a0249517e1885000daf4a020d7d6ce9aae3a76928e9fc74e5beb7cd21224d2c","family_id":"qf-mod37-estvar-at-parameter","seed":202611064008,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"parameter\":1,\"statistic\":[0,1,9,10]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 53/7, 754/49, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 53/7, 754/49, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0009","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"773720210e4174f1c5f64158783777b33e7394faaca23dfe12ac4b8114dbb061","family_id":"qf-mod37-estvar-at-parameter","seed":202611064009,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"parameter\":0,\"statistic\":[0,1,10,11]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 82/9, 1169/81, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 82/9, 1169/81, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0010","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49944ca91b13fdd84b5603d1409e45b3d7e7d434aba53551a4dd02ec045e7b0b","family_id":"qf-mod37-estvar-at-parameter","seed":202611064010,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"parameter\":1,\"statistic\":[0,1,11,12]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 215/23, 11344/529, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 215/23, 11344/529, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0011","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3d47717e97660d1a859a6577458c4172bbe2ba59ca91101aa7443b73e389f400","family_id":"qf-mod37-estvar-at-parameter","seed":202611064011,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"parameter\":0,\"statistic\":[0,1,12,13]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 11, 19, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 11, 19, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0012","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"493d7b7497e095493f5fe14dc68aeb671838e536a633b591d16ba80eb0e5dbc8","family_id":"qf-mod37-estvar-at-parameter","seed":202611064012,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"parameter\":1,\"statistic\":[0,1,13,14]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 279/25, 17534/625, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 279/25, 17534/625, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0013","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cda2d42ecb19fe21fabd61f14f9bf58d05096d29cfe5af167d3022346662fa85","family_id":"qf-mod37-estvar-at-parameter","seed":202611064013,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"parameter\":0,\"statistic\":[0,1,14,15]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 142/11, 2881/121, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 142/11, 2881/121, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0014","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6553704ecb625dbf4cf9274eb5b8578f6d7936c46e36f953a3ac12a30ded868","family_id":"qf-mod37-estvar-at-parameter","seed":202611064014,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"parameter\":1,\"statistic\":[0,1,15,16]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 13, 316/9, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 13, 316/9, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0015","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1ec416200afb288dfa86f1430d0c2296f9afc7615d9c4536e7fb36999096cc67","family_id":"qf-mod37-estvar-at-parameter","seed":202611064015,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"parameter\":0,\"statistic\":[0,1,16,17]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 89/6, 1037/36, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 89/6, 1037/36, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0016","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e5c5f5a2cd3b32dd747e2c67bb0fd0f45c009e368baaeda8b03fb26be33f2ac","family_id":"qf-mod37-estvar-at-parameter","seed":202611064016,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"parameter\":1,\"statistic\":[0,1,17,18]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 431/29, 35770/841, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 431/29, 35770/841, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0017","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49d4e7bac8b31c39e46037a68493e33648a6275b12cdd87180fdd730e14761b9","family_id":"qf-mod37-estvar-at-parameter","seed":202611064017,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"parameter\":0,\"statistic\":[0,1,18,19]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 218/13, 5737/169, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 218/13, 5737/169, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0018","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4311037294dc2aca9d8d299c4b5172e3231143e7e35e59309761f43f659e9b6e","family_id":"qf-mod37-estvar-at-parameter","seed":202611064018,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"parameter\":1,\"statistic\":[0,1,19,20]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 519/31, 48296/961, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 519/31, 48296/961, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0019","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"53b835aa952b0a4794ac6201e065ce97154390cbc61b716c9c69ddbeabff5cee","family_id":"qf-mod37-estvar-at-parameter","seed":202611064019,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"parameter\":0,\"statistic\":[0,1,20,21]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 131/7, 1921/49, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 131/7, 1921/49, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0020","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"17d9681c64375e5beb6922c5d32a70f989818dcf3158a31b8713dd050f12007a","family_id":"qf-mod37-estvar-at-parameter","seed":202611064020,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"parameter\":1,\"statistic\":[0,1,21,22]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 205/11, 7046/121, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 205/11, 7046/121, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0021","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3760359a748d9af7f74efabb829c2c1b76901f0fd89d1e2ebc49121e22d20548","family_id":"qf-mod37-estvar-at-parameter","seed":202611064021,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"parameter\":0,\"statistic\":[0,1,22,23]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 62/3, 401/9, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 62/3, 401/9, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0022","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b16ad9fd66006b5dad81a489a026bc1eebee5465a3fa415b4c66aa8cdc85716","family_id":"qf-mod37-estvar-at-parameter","seed":202611064022,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"parameter\":1,\"statistic\":[0,1,23,24]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 719/35, 81364/1225, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 719/35, 81364/1225, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0023","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"baed54d1622ce8f1130fd039db0c43c94201cad4e74e2eb57fee316df106aef5","family_id":"qf-mod37-estvar-at-parameter","seed":202611064023,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"parameter\":0,\"statistic\":[0,1,24,25]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 181/8, 3199/64, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 181/8, 3199/64, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0024","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"00281841a7abc1eac98aa03759898c2a39a8576ba8f136e719efc69054da2014","family_id":"qf-mod37-estvar-at-parameter","seed":202611064024,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"parameter\":1,\"statistic\":[0,1,25,26]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(1, 831/37, 102386/1369, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 831/37, 102386/1369, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0025","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b69a1740acb7c5bb89296d3a2a1558beadd594ce0100851d4e3d78dd5fa21bad","family_id":"qf-mod37-estvar-at-parameter","seed":202611064025,"semantic_template":"qf-mod37-estvar-at-parameter/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"parameter\":0,\"statistic\":[0,1,26,27]}"},"fields":{"title":"Use One Parameter Law and Its Own Mean for Variance","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).","answer":{"kind":"expression","value":"(0, 418/17, 16033/289, Yes, same_theta_sampling_law_support_and_mean)","explanation":"Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 418/17, 16033/289, Yes, same_theta_sampling_law_support_and_mean)."},"skills":["Use One Parameter Law and Its Own Mean for Variance"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0026","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1a28820a66de91305e11deca1eb45e569d150d7f46d9626a696c43cf5113639d","family_id":"qf-mod37-estvar-family-readback","seed":202611064026,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 2, 1], [1, 13/7, 55/49]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 2, 1], [1, 13/7, 55/49]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0027","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0db96a9ddf774168ef6880b0639530cbc6c849447896d894318d40c487fde3eb","family_id":"qf-mod37-estvar-family-readback","seed":202611064027,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0028","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"991a25f5c19ee8f4048e2ef21024ee8c622dddfb2fc34b9410e2b506ede3059d","family_id":"qf-mod37-estvar-family-readback","seed":202611064028,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 11/3, 29/9], [1, 27/8, 239/64]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 11/3, 29/9], [1, 27/8, 239/64]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0029","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42e52100f22ca71bbdaf4c2e52e8fcd0839b978f6437fd42831a2a68594935d3","family_id":"qf-mod37-estvar-family-readback","seed":202611064029,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 59/13, 796/169], [1, 71/17, 1606/289]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 59/13, 796/169], [1, 71/17, 1606/289]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0030","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a02aabfbf8fc94668266dc35d5c9207c96f943f01c2b7f3068d5495b131796c3","family_id":"qf-mod37-estvar-family-readback","seed":202611064030,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0031","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1b04878edd307d227a7562777201fa3aca96b71576ae73c160a58fa062acea6e","family_id":"qf-mod37-estvar-family-readback","seed":202611064031,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 19/3, 74/9], [1, 111/19, 3620/361]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 19/3, 74/9], [1, 111/19, 3620/361]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0032","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf42ea1e3892a7ae27e4df6e40643216611082200bfe5e5ce6b263bda6a51a90","family_id":"qf-mod37-estvar-family-readback","seed":202611064032,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 29/4, 163/16], [1, 67/10, 1261/100]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 29/4, 163/16], [1, 67/10, 1261/100]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0033","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"942289904870ea0bc82fc335b82317befe7467e52da7769c05f50edb0b0278e9","family_id":"qf-mod37-estvar-family-readback","seed":202611064033,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 139/17, 3544/289], [1, 53/7, 754/49]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 139/17, 3544/289], [1, 53/7, 754/49]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0034","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9d1ec3c5200dd1439c9db273fb67b5aca953150b7cebeb935331ab3aceb8a370","family_id":"qf-mod37-estvar-family-readback","seed":202611064034,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 82/9, 1169/81], [1, 93/11, 2219/121]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 82/9, 1169/81], [1, 93/11, 2219/121]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0035","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"029fa0e858e3b48073a4e325c7fe12cc9dea0fe0398db81343e0efffe909ab62","family_id":"qf-mod37-estvar-family-readback","seed":202611064035,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 191/19, 6022/361], [1, 215/23, 11344/529]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 191/19, 6022/361], [1, 215/23, 11344/529]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0036","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2f51d2ba238fdaa8f9d22c7f49cfe16a96aa94691a4b8cfbf05d4ebcd6f4c18","family_id":"qf-mod37-estvar-family-readback","seed":202611064036,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 11, 19], [1, 41/4, 395/16]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 11, 19], [1, 41/4, 395/16]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0037","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7716bb9a95c02e2a8e1fc7fbb8d2bd5ee7a86b38d9ae5add856558b52d14ab0a","family_id":"qf-mod37-estvar-family-readback","seed":202611064037,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 251/21, 9428/441], [1, 279/25, 17534/625]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 251/21, 9428/441], [1, 279/25, 17534/625]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0038","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8006cfe0877563b0147f89836654177e064d9d8fcdaf78a0b09de12381a15a67","family_id":"qf-mod37-estvar-family-readback","seed":202611064038,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 142/11, 2881/121], [1, 157/13, 5329/169]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 142/11, 2881/121], [1, 157/13, 5329/169]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0039","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"42a7c63ec2d83b80e7be6df449761db50203d654d29da20f6ca0802fc865789f","family_id":"qf-mod37-estvar-family-readback","seed":202611064039,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0040","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c503cf2e8570bb9e38775824dba1fc2b075387930a7c45960754f8fc1a49b2f","family_id":"qf-mod37-estvar-family-readback","seed":202611064040,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 89/6, 1037/36], [1, 195/14, 7601/196]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 89/6, 1037/36], [1, 195/14, 7601/196]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0041","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8bef2173cf4d3aba6cccf28b385509e50889b74b774bbbb3b64f3380e948dfab","family_id":"qf-mod37-estvar-family-readback","seed":202611064041,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 79/5, 784/25], [1, 431/29, 35770/841]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 79/5, 784/25], [1, 431/29, 35770/841]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0042","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0cf53ad3405ff8dc8f631f43f70ecbac5d3acff710da864199a1d191ec118dec","family_id":"qf-mod37-estvar-family-readback","seed":202611064042,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 218/13, 5737/169], [1, 79/5, 1159/25]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 218/13, 5737/169], [1, 79/5, 1159/25]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0043","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bd1065044b6958d1c28413a683312f6f056312915e09e0f707c108a026750a5","family_id":"qf-mod37-estvar-family-readback","seed":202611064043,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 479/27, 26654/729], [1, 519/31, 48296/961]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 479/27, 26654/729], [1, 519/31, 48296/961]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0044","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f725e5d6c61ca1d0742ec4a254b4e213f9d104fb0f41b0f6e44813f5d5640a6c","family_id":"qf-mod37-estvar-family-readback","seed":202611064044,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 131/7, 1921/49], [1, 283/16, 13879/256]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 131/7, 1921/49], [1, 283/16, 13879/256]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0045","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97a75400352ddaf1ac1f63ccf3988d77c5b1fa53869742bddf35e49658f57362","family_id":"qf-mod37-estvar-family-readback","seed":202611064045,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 571/29, 35212/841], [1, 205/11, 7046/121]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 571/29, 35212/841], [1, 205/11, 7046/121]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0046","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea5cb97cea2b041f7603b1c14f258e2da0a01e13a2ad2d72966ba58b805424f9","family_id":"qf-mod37-estvar-family-readback","seed":202611064046,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 62/3, 401/9], [1, 333/17, 18005/289]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 62/3, 401/9], [1, 333/17, 18005/289]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0047","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a9b1b75eb22b757129265ce5e66f41fa8fb4b7c6deefe58e2cb4cffc5f51fabe","family_id":"qf-mod37-estvar-family-readback","seed":202611064047,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 671/31, 45418/961], [1, 719/35, 81364/1225]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 671/31, 45418/961], [1, 719/35, 81364/1225]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0048","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8248f2650af6e6583ea2cf1d4cf72f42a5e30c450430a29abaa2dbccf43abaff","family_id":"qf-mod37-estvar-family-readback","seed":202611064048,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 181/8, 3199/64], [1, 43/2, 847/12]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 181/8, 3199/64], [1, 43/2, 847/12]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0049","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8669ea0e0f2dee2f24da2929a1033f0ccebe03f8c0d51206123ed719888eee31","family_id":"qf-mod37-estvar-family-readback","seed":202611064049,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0050","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"45dbe5afacab212578adeeb42d2703d630e5fcd609e6e29fcfbfbece01b80e3e","family_id":"qf-mod37-estvar-family-readback","seed":202611064050,"semantic_template":"qf-mod37-estvar-family-readback/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27]}"},"fields":{"title":"Estimator Variance Remains Parameter-Indexed","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).","answer":{"kind":"expression","value":"([[0, 418/17, 16033/289], [1, 445/19, 28531/361]], variance_is_parameter_indexed_family)","explanation":"For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 418/17, 16033/289], [1, 445/19, 28531/361]], variance_is_parameter_indexed_family)."},"skills":["Estimator Variance Remains Parameter-Indexed"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0051","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"23b2ed794f1a5a76bb8ec5aca0138cfdba442f2cd929fd9825948f5ce7169273","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064051,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"parameter\":0,\"statistic\":[0,1,2,3]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 5, 2, 1, 1, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 5, 2, 1, 1, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0052","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8d966e1c65f658a236bf547dbd24d8877a92ba97eab50dcc977783a10be8f5f5","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064052,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"parameter\":1,\"statistic\":[0,1,3,4]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 9, 13/5, 56/25, 56/25, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 9, 13/5, 56/25, 56/25, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0053","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2503cc719907dd7675bc195486ac0b4a80eb73388f5229eff83817d914fc37d8","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064053,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"parameter\":0,\"statistic\":[0,1,4,5]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 50/3, 11/3, 29/9, 29/9, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 50/3, 11/3, 29/9, 29/9, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0054","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0c40a9a0d0ad24acf0e00e787e6759f0c06c799a4bc3dace0b39a3395dfc4e49","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064054,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"parameter\":1,\"statistic\":[0,1,5,6]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 23, 71/17, 1606/289, 1606/289, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 23, 71/17, 1606/289, 1606/289, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0055","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"833d514514985181bcb9e422e0a63987d8426411e72e6c596ec6f80d2f2de85c","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064055,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"parameter\":0,\"statistic\":[0,1,6,7]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 251/7, 38/7, 313/49, 313/49, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 251/7, 38/7, 313/49, 313/49, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0056","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dd9afe8888f64f944c99d16b2cbd2f62093cc1a87f7becb7bb1776809d181a95","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064056,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"parameter\":1,\"statistic\":[0,1,7,8]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 839/19, 111/19, 3620/361, 3620/361, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 839/19, 111/19, 3620/361, 3620/361, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0057","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ea6b4ce57f43aa3d2e6c5d0d637553a9bca9b8dc30158fa19d1e8cb60423f3ff","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064057,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"parameter\":0,\"statistic\":[0,1,8,9]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 251/4, 29/4, 163/16, 163/16, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 251/4, 29/4, 163/16, 163/16, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0058","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"97486817cc4c244a27eafe6ded87d046f2dcd01b2b0c0d1ff7c502d12fa9a168","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064058,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"parameter\":1,\"statistic\":[0,1,9,10]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 509/7, 53/7, 754/49, 754/49, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 509/7, 53/7, 754/49, 754/49, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0059","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"af8c3141ca2d6428d1ebfc627eee4a4cd19d87777939d061e169630f8ae3aebf","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064059,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"parameter\":0,\"statistic\":[0,1,10,11]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 877/9, 82/9, 1169/81, 1169/81, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 877/9, 82/9, 1169/81, 1169/81, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0060","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f8c2b7bf70aefc1c886e8ae76a92932fde7de7e95855422030ad3bdb8c8ef47d","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064060,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"parameter\":1,\"statistic\":[0,1,11,12]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 2503/23, 215/23, 11344/529, 11344/529, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 2503/23, 215/23, 11344/529, 11344/529, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0061","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"701a3a2263f83cb6888d842ea2a0f896ea37d7ecfbc9add4eb2b53aadd73aebc","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064061,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"parameter\":0,\"statistic\":[0,1,12,13]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 140, 11, 19, 19, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 140, 11, 19, 19, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0062","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cf4ac84bd10c31dbe84780f247e3ada9a92a43f14b09194826aaf34295c6eba6","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064062,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"parameter\":1,\"statistic\":[0,1,13,14]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 763/5, 279/25, 17534/625, 17534/625, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 763/5, 279/25, 17534/625, 17534/625, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0063","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"47ec7314e98a3852b761c34f67603184b5baa14de897a1b0dd5eea72837e43f7","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064063,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"parameter\":0,\"statistic\":[0,1,14,15]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 2095/11, 142/11, 2881/121, 2881/121, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 2095/11, 142/11, 2881/121, 2881/121, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0064","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b77399819b8d0abef78feced40e846586cf4ddece67216f9ed641e300e983466","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064064,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"parameter\":1,\"statistic\":[0,1,15,16]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 1837/9, 13, 316/9, 316/9, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 1837/9, 13, 316/9, 316/9, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0065","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"774d422bd18e95dd96038ae89190d13360eb98a827327abd32afb4b1587d0ee1","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064065,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"parameter\":0,\"statistic\":[0,1,16,17]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 1493/6, 89/6, 1037/36, 1037/36, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 1493/6, 89/6, 1037/36, 1037/36, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0066","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f49756265a457d0dd66e28b0240d647aa8d66d927a8da1e945232e8118732734","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064066,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"parameter\":1,\"statistic\":[0,1,17,18]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 7639/29, 431/29, 35770/841, 35770/841, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 7639/29, 431/29, 35770/841, 35770/841, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0067","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cb3fbf26ed615c87fd867ae086c154f089b5887eef28c35390bfd07a1cd6180b","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064067,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"parameter\":0,\"statistic\":[0,1,18,19]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 4097/13, 218/13, 5737/169, 5737/169, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 4097/13, 218/13, 5737/169, 5737/169, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0068","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"93f0841777fd79a8be0132ffc64ec9cadf6c27ebeac4e27324858f201c91406c","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064068,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"parameter\":1,\"statistic\":[0,1,19,20]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 10247/31, 519/31, 48296/961, 48296/961, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 10247/31, 519/31, 48296/961, 48296/961, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0069","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b2e220e30e7fe133138c16b4b6afbedd6f61db196fb329291eb4c7aa41dd3fcf","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064069,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"parameter\":0,\"statistic\":[0,1,20,21]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 2726/7, 131/7, 1921/49, 1921/49, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 2726/7, 131/7, 1921/49, 1921/49, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0070","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3df262b59b0992be8d6330a318c51b89faaa05179ad1f9748b286aa9a5824184","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064070,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"parameter\":1,\"statistic\":[0,1,21,22]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 4461/11, 205/11, 7046/121, 7046/121, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 4461/11, 205/11, 7046/121, 7046/121, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0071","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9660659095cdf093078ab9690e805de89399588698ef150067bf3e75ad2cac96","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064071,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"parameter\":0,\"statistic\":[0,1,22,23]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 1415/3, 62/3, 401/9, 401/9, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 1415/3, 62/3, 401/9, 401/9, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0072","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ba4830e327c314d16d0e9a73eb03e26230b807a9f697b7c48d472c794bd7d4b3","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064072,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"parameter\":1,\"statistic\":[0,1,23,24]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 3419/7, 719/35, 81364/1225, 81364/1225, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 3419/7, 719/35, 81364/1225, 81364/1225, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0073","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bcb9f7488cd4712d2c7c32a45e0c4194b369b5f3a960b21ffa8aa5ff5ab5772e","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064073,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"parameter\":0,\"statistic\":[0,1,24,25]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 4495/8, 181/8, 3199/64, 3199/64, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 4495/8, 181/8, 3199/64, 3199/64, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0074","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5dcb83be322d884e7ca14cb2c568895ce9810a2952de78a87e6587197e586efe","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064074,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"parameter\":1,\"statistic\":[0,1,25,26]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(1, 21431/37, 831/37, 102386/1369, 102386/1369, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 21431/37, 831/37, 102386/1369, 102386/1369, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0075","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dadc9209ce41832ceb4d29635155d4df79454579c3a20ce1a3b46799eea8cf1e","family_id":"qf-mod37-estvar-second-moment-identity","seed":202611064075,"semantic_template":"qf-mod37-estvar-second-moment-identity/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"parameter\":0,\"statistic\":[0,1,26,27]}"},"fields":{"title":"The Second Moment and Squared Mean Are Different Quantities","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).","answer":{"kind":"expression","value":"(0, 11221/17, 418/17, 16033/289, 16033/289, Yes)","explanation":"Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 11221/17, 418/17, 16033/289, 16033/289, Yes)."},"skills":["The Second Moment and Squared Mean Are Different Quantities"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0076","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c002f91cb88f75138dd4eb5cdee89dd6079715e69c519490fe063c52d0128abe","family_id":"qf-mod37-estvar-property-boundary","seed":202611064076,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,10],[2,10],[3,10],[4,10]],[[2,14],[3,14],[4,14],[5,14]]],\"statistic\":[0,1,2,3]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 1], [1, 55/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1], [1, 55/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0077","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"59007b153964da166fb478e9746d23a4bd989e2346e576b9da927cd6c930cd19","family_id":"qf-mod37-estvar-property-boundary","seed":202611064077,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,11],[2,11],[3,11],[5,11]],[[2,15],[3,15],[4,15],[6,15]]],\"statistic\":[0,1,3,4]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 238/121], [1, 56/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 238/121], [1, 56/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0078","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e30b88eb1ca3cba6a5b96d5f43273f191bb40aaf0c00a857c6e32add9677406d","family_id":"qf-mod37-estvar-property-boundary","seed":202611064078,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,12],[2,12],[3,12],[6,12]],[[2,16],[3,16],[4,16],[7,16]]],\"statistic\":[0,1,4,5]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 29/9], [1, 239/64]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 29/9], [1, 239/64]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0079","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9458598a309ed0c6d5deec1d1442a7b9cd4d2dcafd6788d7bbf83a335eb5d645","family_id":"qf-mod37-estvar-property-boundary","seed":202611064079,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,13],[2,13],[3,13],[7,13]],[[2,17],[3,17],[4,17],[8,17]]],\"statistic\":[0,1,5,6]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 796/169], [1, 1606/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 796/169], [1, 1606/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0080","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4f793337f6cccaf5903f49164dbe19630b7f4ee525e35508f6c1a3159d56079d","family_id":"qf-mod37-estvar-property-boundary","seed":202611064080,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,14],[2,14],[3,14],[8,14]],[[2,18],[3,18],[4,18],[9,18]]],\"statistic\":[0,1,6,7]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 313/49], [1, 23/3]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 313/49], [1, 23/3]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0081","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fc55cfcd2d8f127104e79353c53853a6ba7e236472e47df7caf6e453727fa757","family_id":"qf-mod37-estvar-property-boundary","seed":202611064081,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,15],[2,15],[3,15],[9,15]],[[2,19],[3,19],[4,19],[10,19]]],\"statistic\":[0,1,7,8]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 74/9], [1, 3620/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 74/9], [1, 3620/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0082","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"095ee9ff277d938374a6dc76177d26f6c48608551f1a3453a921d8565db23cc5","family_id":"qf-mod37-estvar-property-boundary","seed":202611064082,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,16],[2,16],[3,16],[10,16]],[[2,20],[3,20],[4,20],[11,20]]],\"statistic\":[0,1,8,9]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 163/16], [1, 1261/100]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 163/16], [1, 1261/100]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0083","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7bcd3c29156654773d4c193ad954e11516d06f6cccf810b7a03501122de57569","family_id":"qf-mod37-estvar-property-boundary","seed":202611064083,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,17],[2,17],[3,17],[11,17]],[[2,21],[3,21],[4,21],[12,21]]],\"statistic\":[0,1,9,10]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 3544/289], [1, 754/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 3544/289], [1, 754/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0084","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5c3282111ba52268e7fe7d034585bc1feca96cf6a1665f1667d1d7d55754c58f","family_id":"qf-mod37-estvar-property-boundary","seed":202611064084,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,18],[2,18],[3,18],[12,18]],[[2,22],[3,22],[4,22],[13,22]]],\"statistic\":[0,1,10,11]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 1169/81], [1, 2219/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1169/81], [1, 2219/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0085","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5d0f999fb68e41a9a9fe0567f380359bb46039ae58d8f8a4cee9c447d0178932","family_id":"qf-mod37-estvar-property-boundary","seed":202611064085,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,19],[2,19],[3,19],[13,19]],[[2,23],[3,23],[4,23],[14,23]]],\"statistic\":[0,1,11,12]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 6022/361], [1, 11344/529]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 6022/361], [1, 11344/529]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0086","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9860eff9f35af6b76ee2d674858e3632c4129165f67d9a34ce8892cbb8a8b53c","family_id":"qf-mod37-estvar-property-boundary","seed":202611064086,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,20],[2,20],[3,20],[14,20]],[[2,24],[3,24],[4,24],[15,24]]],\"statistic\":[0,1,12,13]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 19], [1, 395/16]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 19], [1, 395/16]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0087","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6f1e5be582fe485edf1b32708572c3b44a4e33c496ee073ff08546beb70cff38","family_id":"qf-mod37-estvar-property-boundary","seed":202611064087,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,21],[2,21],[3,21],[15,21]],[[2,25],[3,25],[4,25],[16,25]]],\"statistic\":[0,1,13,14]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 9428/441], [1, 17534/625]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 9428/441], [1, 17534/625]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0088","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"51311f2c4af20cd02ce5a3fd48e66f7921f1cd84227e098a43f4382d06490420","family_id":"qf-mod37-estvar-property-boundary","seed":202611064088,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,22],[2,22],[3,22],[16,22]],[[2,26],[3,26],[4,26],[17,26]]],\"statistic\":[0,1,14,15]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 2881/121], [1, 5329/169]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 2881/121], [1, 5329/169]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0089","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2b68abb9a2eb8495ea37d5251f6c3c39880ea839dce4ff0f8b22f7963d113e7e","family_id":"qf-mod37-estvar-property-boundary","seed":202611064089,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,23],[2,23],[3,23],[17,23]],[[2,27],[3,27],[4,27],[18,27]]],\"statistic\":[0,1,15,16]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 13906/529], [1, 316/9]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 13906/529], [1, 316/9]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0090","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e0cdc3032103da30cf0d856e86c535b90139bce1646bdd4e8a49c65dad9d0908","family_id":"qf-mod37-estvar-property-boundary","seed":202611064090,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,24],[2,24],[3,24],[18,24]],[[2,28],[3,28],[4,28],[19,28]]],\"statistic\":[0,1,16,17]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 1037/36], [1, 7601/196]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1037/36], [1, 7601/196]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0091","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b080b8a1614d81df300a94e5643e20dfc3e8d52de1858144796423678c69421","family_id":"qf-mod37-estvar-property-boundary","seed":202611064091,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,25],[2,25],[3,25],[19,25]],[[2,29],[3,29],[4,29],[20,29]]],\"statistic\":[0,1,17,18]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 784/25], [1, 35770/841]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 784/25], [1, 35770/841]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0092","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"286b554b68baecdb9fbcdc595b641b22f38046066633db9665a800d94d499cc6","family_id":"qf-mod37-estvar-property-boundary","seed":202611064092,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,26],[2,26],[3,26],[20,26]],[[2,30],[3,30],[4,30],[21,30]]],\"statistic\":[0,1,18,19]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 5737/169], [1, 1159/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 5737/169], [1, 1159/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0093","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9837ce6f8f8cd9c38c0ec652083b01e45bfba0b8091bd21b2e405f7ff7970730","family_id":"qf-mod37-estvar-property-boundary","seed":202611064093,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,27],[2,27],[3,27],[21,27]],[[2,31],[3,31],[4,31],[22,31]]],\"statistic\":[0,1,19,20]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 26654/729], [1, 48296/961]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 26654/729], [1, 48296/961]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0094","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a61aa3fb2ece8f5a924f390ed781672f390dcb8682e1105a63cdad096f76140f","family_id":"qf-mod37-estvar-property-boundary","seed":202611064094,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,28],[2,28],[3,28],[22,28]],[[2,32],[3,32],[4,32],[23,32]]],\"statistic\":[0,1,20,21]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 1921/49], [1, 13879/256]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1921/49], [1, 13879/256]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0095","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05d1b194c14086922c4d0f9ffbaff9eaab344e23d205c0f0698db73a8804c17b","family_id":"qf-mod37-estvar-property-boundary","seed":202611064095,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,29],[2,29],[3,29],[23,29]],[[2,33],[3,33],[4,33],[24,33]]],\"statistic\":[0,1,21,22]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 35212/841], [1, 7046/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 35212/841], [1, 7046/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0096","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b94ef935defa083166b43ffd8f40a26e57cb79e669415df9fdb0dc3e555fd6b3","family_id":"qf-mod37-estvar-property-boundary","seed":202611064096,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,30],[2,30],[3,30],[24,30]],[[2,34],[3,34],[4,34],[25,34]]],\"statistic\":[0,1,22,23]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 401/9], [1, 18005/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 401/9], [1, 18005/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0097","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d8ce792a3e2aa89b3ea22ed3f9e11fc3bd4b672e1223d5d905e3651fd5e0517f","family_id":"qf-mod37-estvar-property-boundary","seed":202611064097,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,31],[2,31],[3,31],[25,31]],[[2,35],[3,35],[4,35],[26,35]]],\"statistic\":[0,1,23,24]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 45418/961], [1, 81364/1225]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 45418/961], [1, 81364/1225]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0098","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"56b12314036ed63698118e15d54a6f81fc222b0f0fdfe95ed3b40217375b22e5","family_id":"qf-mod37-estvar-property-boundary","seed":202611064098,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,32],[2,32],[3,32],[26,32]],[[2,36],[3,36],[4,36],[27,36]]],\"statistic\":[0,1,24,25]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 3199/64], [1, 847/12]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 3199/64], [1, 847/12]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0099","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b5df0d2bf740862ce582dff67b7e27ea1fb51e18ff6a83d5411817802507424a","family_id":"qf-mod37-estvar-property-boundary","seed":202611064099,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,33],[2,33],[3,33],[27,33]],[[2,37],[3,37],[4,37],[28,37]]],\"statistic\":[0,1,25,26]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 57416/1089], [1, 102386/1369]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 57416/1089], [1, 102386/1369]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the 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-20261106-r04-estimator-variance-0100","batch_id":"batch-20261014-advance-20261106-r04-estimator-variance-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fe2f726915a91bdb4e1e2ea1f3bb9d6bbf0f4dc4318810f17ceea852a43cc854","family_id":"qf-mod37-estvar-property-boundary","seed":202611064100,"semantic_template":"qf-mod37-estvar-property-boundary/{\"laws\":[[[1,34],[2,34],[3,34],[28,34]],[[2,38],[3,38],[4,38],[29,38]]],\"statistic\":[0,1,26,27]}"},"fields":{"title":"Variance Alone Withholds Unbiasedness, Consistency and MSE Claims","stem":"Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).","answer":{"kind":"expression","value":"([[0, 16033/289], [1, 28531/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)","explanation":"List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 16033/289], [1, 28531/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)."},"skills":["Variance Alone Withholds Unbiasedness, Consistency and MSE Claims"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}