{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0001","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3cbeb359a16acad6a19176f930b48c25cabe16b7bc826b3605023d657ac5bf56","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145001,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-11,1],[-12,1],[-13,1]],\"theta\":[-12,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-12, complete listed estimator values=[-13, -12, -11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-12, [[-13, 1/4, 1, 1/4], [-12, 1/2, 0, 0], [-11, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-12, [[-13, 1/4, 1, 1/4], [-12, 1/2, 0, 0], [-11, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0002","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ef6b755756ba88868e0026eeef7c6fd95dc4528b49874a85c6c754b2a133769f","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145002,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-11,1],[-13,1],[-9,1]],\"theta\":[-11,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-11, complete listed estimator values=[-13, -11, -9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-11, [[-13, 1/4, 4, 1], [-11, 1/2, 0, 0], [-9, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-11, [[-13, 1/4, 4, 1], [-11, 1/2, 0, 0], [-9, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0003","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"10a010e1e44e9906c92a777cf4b84643c84f5dcdf6d6f64eb8cc3e32042a37aa","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145003,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-10, complete listed estimator values=[-13, -10, -7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-10, [[-13, 1/4, 9, 9/4], [-10, 1/2, 0, 0], [-7, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0004","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dda18fa5f85f2f8fab142c537e6b20f13d9fd83a94e83b3e3e503958444278a1","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145004,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-5,1],[-9,1]],\"theta\":[-9,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-9, complete listed estimator values=[-13, -9, -5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-9, [[-13, 1/4, 16, 4], [-9, 1/2, 0, 0], [-5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-9, [[-13, 1/4, 16, 4], [-9, 1/2, 0, 0], [-5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0005","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92c1f94034875709d198ea803a6855e8348dea98873c33a03b6a07d7334162b8","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145005,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-3,1],[-8,1]],\"theta\":[-8,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-8, complete listed estimator values=[-13, -8, -3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-8, [[-13, 1/4, 25, 25/4], [-8, 1/2, 0, 0], [-3, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-8, [[-13, 1/4, 25, 25/4], [-8, 1/2, 0, 0], [-3, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0006","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"86113d485d4cd52558169578bc18fd2938a4440d5e2d87a3c2a0922c0fe14b41","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145006,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-6,1],[-7,1],[-8,1]],\"theta\":[-7,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-7, complete listed estimator values=[-8, -7, -6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-7, [[-8, 1/4, 1, 1/4], [-7, 1/2, 0, 0], [-6, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-7, [[-8, 1/4, 1, 1/4], [-7, 1/2, 0, 0], [-6, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0007","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f41937a99908166ce561f015484bc8707006221ce5072eb042cdfc1f55b193cb","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145007,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-6,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-6, complete listed estimator values=[-8, -6, -4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-6, [[-8, 1/4, 4, 1], [-6, 1/2, 0, 0], [-4, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0008","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e03e0ac3d412107767f7fde5d1903bd970652d5212d503069f0fc4c118d4ae4c","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145008,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-2,1],[-5,1],[-8,1]],\"theta\":[-5,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-5, complete listed estimator values=[-8, -5, -2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-5, [[-8, 1/4, 9, 9/4], [-5, 1/2, 0, 0], [-2, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-5, [[-8, 1/4, 9, 9/4], [-5, 1/2, 0, 0], [-2, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0009","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"1936df3bbd48aeb914349b4b6b93ab626ea83ba0c1a80baf8786a32b1f18eea0","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145009,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-4,1],[-8,1],[0,1]],\"theta\":[-4,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-4, complete listed estimator values=[-8, -4, 0] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-4, [[-8, 1/4, 16, 4], [-4, 1/2, 0, 0], [0, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-4, [[-8, 1/4, 16, 4], [-4, 1/2, 0, 0], [0, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0010","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"684b1ce3ff792077ca86a1cbaaf6d1920a9cfaf0dcd822ce4f818bcba86e0e7c","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145010,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-3,1],[-8,1],[2,1]],\"theta\":[-3,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-3, complete listed estimator values=[-8, -3, 2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-3, [[-8, 1/4, 25, 25/4], [-3, 1/2, 0, 0], [2, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-3, [[-8, 1/4, 25, 25/4], [-3, 1/2, 0, 0], [2, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0011","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4bb1a9df1c8014a74a0fd18907d814be80efe3da621bb1b6da0b8771bfd03879","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145011,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-2,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-2, complete listed estimator values=[-3, -2, -1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-2, [[-3, 1/4, 1, 1/4], [-2, 1/2, 0, 0], [-1, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0012","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"70662953f78a59dc419b2d80f0c7d6e0a369c1bd02970c4d959c0fac767a7265","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145012,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-1,1],[-3,1],[1,1]],\"theta\":[-1,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=-1, complete listed estimator values=[-3, -1, 1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(-1, [[-3, 1/4, 4, 1], [-1, 1/2, 0, 0], [1, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (-1, [[-3, 1/4, 4, 1], [-1, 1/2, 0, 0], [1, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0013","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7e315e7001180b2c8a773d22079316e5f937f60674e5c769342087b6e91c9e5f","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145013,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[0,1],[3,1]],\"theta\":[0,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=0, complete listed estimator values=[-3, 0, 3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (0, [[-3, 1/4, 9, 9/4], [0, 1/2, 0, 0], [3, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0014","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"02503ebc9c8c19abbe3bc01c2e9bcd46c2254fbb7b881e225f270274d390cc9d","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145014,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[1,1],[5,1]],\"theta\":[1,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=1, complete listed estimator values=[-3, 1, 5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(1, [[-3, 1/4, 16, 4], [1, 1/2, 0, 0], [5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (1, [[-3, 1/4, 16, 4], [1, 1/2, 0, 0], [5, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0015","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c26ea9e5fe6b90df0347f21c5eab0ae268eaa9371bae5cdcf77171d59241bc71","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145015,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[2,1],[7,1]],\"theta\":[2,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=2, complete listed estimator values=[-3, 2, 7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(2, [[-3, 1/4, 25, 25/4], [2, 1/2, 0, 0], [7, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (2, [[-3, 1/4, 25, 25/4], [2, 1/2, 0, 0], [7, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0016","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"138a142f72006d3b5f9c54d9d57262eb23a4fc99a7375870738233fd42df39dc","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145016,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[3,1],[4,1]],\"theta\":[3,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=3, complete listed estimator values=[2, 3, 4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (3, [[2, 1/4, 1, 1/4], [3, 1/2, 0, 0], [4, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0017","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0824e135608d500f719d6d87b7a6ea9683b0c0d8a8f3f4ac5115e796a8d20410","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145017,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[4,1],[6,1]],\"theta\":[4,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=4, complete listed estimator values=[2, 4, 6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(4, [[2, 1/4, 4, 1], [4, 1/2, 0, 0], [6, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (4, [[2, 1/4, 4, 1], [4, 1/2, 0, 0], [6, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0018","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e8aae18fb310a286eba8b2fb81134b11b2782c930f044c199d754d2c94321c6a","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145018,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[5,1],[8,1]],\"theta\":[5,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=5, complete listed estimator values=[2, 5, 8] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(5, [[2, 1/4, 9, 9/4], [5, 1/2, 0, 0], [8, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (5, [[2, 1/4, 9, 9/4], [5, 1/2, 0, 0], [8, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0019","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9de6860b0a5205d4ea296d528131ce49c4ca2bd41f50094b4ca030d713ff33af","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145019,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[10,1],[2,1],[6,1]],\"theta\":[6,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=6, complete listed estimator values=[2, 6, 10] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(6, [[2, 1/4, 16, 4], [6, 1/2, 0, 0], [10, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (6, [[2, 1/4, 16, 4], [6, 1/2, 0, 0], [10, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0020","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"50e8e07494fcf0679872882b1a3920191e5350dbb49a10246ba6bf7877d95338","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145020,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[12,1],[2,1],[7,1]],\"theta\":[7,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=7, complete listed estimator values=[2, 7, 12] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(7, [[2, 1/4, 25, 25/4], [7, 1/2, 0, 0], [12, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (7, [[2, 1/4, 25, 25/4], [7, 1/2, 0, 0], [12, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0021","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e4176da28dec4e3daf3cedd4ed61004daae07abdbd444a3013cbd74092913b82","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145021,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[7,1],[8,1],[9,1]],\"theta\":[8,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=8, complete listed estimator values=[7, 8, 9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(8, [[7, 1/4, 1, 1/4], [8, 1/2, 0, 0], [9, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (8, [[7, 1/4, 1, 1/4], [8, 1/2, 0, 0], [9, 1/4, 1, 1/4]], 1/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0022","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"32371e018f1eaa7ca7fc6ffe5d37527065dcb291e45345731de346a162814bbd","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145022,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[7,1],[9,1]],\"theta\":[9,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=9, complete listed estimator values=[7, 9, 11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (9, [[7, 1/4, 4, 1], [9, 1/2, 0, 0], [11, 1/4, 4, 1]], 2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0023","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a7cec4bcac2e3600bc027d03171718028a1487ee27e6f9484050e9bd707d558d","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145023,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[13,1],[7,1]],\"theta\":[10,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=10, complete listed estimator values=[7, 10, 13] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(10, [[7, 1/4, 9, 9/4], [10, 1/2, 0, 0], [13, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (10, [[7, 1/4, 9, 9/4], [10, 1/2, 0, 0], [13, 1/4, 9, 9/4]], 9/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0024","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"222d691d42c639b6d4979227ec347f6ba9cfa21c18af7f860347f80db8789649","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145024,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[11,1],[15,1],[7,1]],\"theta\":[11,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=11, complete listed estimator values=[7, 11, 15] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(11, [[7, 1/4, 16, 4], [11, 1/2, 0, 0], [15, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (11, [[7, 1/4, 16, 4], [11, 1/2, 0, 0], [15, 1/4, 16, 4]], 8, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0025","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"01b62409e2558d165fee64cc6dea3b3ca894946649676b7e9ad4a9511c1752f2","family_id":"qf-mod45-mse-single-theta-exact","seed":202611145025,"semantic_template":"qf-mod45-mse-single-theta-exact/{\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[12,1],[17,1],[7,1]],\"theta\":[12,1]}"},"fields":{"title":"MSE Is the Complete Expected Squared Error under One Theta Law","stem":"Use scalar theta=12, complete listed estimator values=[7, 12, 17] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, certificate).","answer":{"kind":"expression","value":"(12, [[7, 1/4, 25, 25/4], [12, 1/2, 0, 0], [17, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)","explanation":"Each output row is [estimator value,probability,squared error,weighted squared error]. Sum every weighted loss directly. Do not replace the definition with bias-variance decomposition, one realized loss or a variance computed under another law. The exact result, retaining the requested order, is (12, [[7, 1/4, 25, 25/4], [12, 1/2, 0, 0], [17, 1/4, 25, 25/4]], 25/2, direct_expected_squared_error_without_bias_variance_decomposition)."},"skills":["MSE Is the Complete Expected Squared Error under One Theta Law"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0026","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b0f2301fb4d57a9d8562dab8b0d4f48f8cd20728d54ac7335fe366200c46758c","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145026,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-11,1],[[-10,1],[-11,1],[-12,1]],[[1,4],[1,2],[1,4]]],[[-12,1],[[-11,1],[-12,1],[-13,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-12, [-13, -12, -11], [[1, 4], [1, 2], [1, 4]]], [-11, [-12, -11, -10], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-11, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], 1/2], [-12, [[-13, 1/4], [-12, 1/2], [-11, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-11, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], 1/2], [-12, [[-13, 1/4], [-12, 1/2], [-11, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0027","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4dcd6c309de67c60c2a682101694fade3ec9b150a16e47a0ed9444822cb41384","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145027,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-10,1],[[-10,1],[-11,1],[-9,1]],[[1,2],[1,4],[1,4]]],[[-11,1],[[-11,1],[-13,1],[-9,1]],[[1,2],[1,4],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-11, [-13, -11, -9], [[1, 4], [1, 2], [1, 4]]], [-10, [-11, -10, -9], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-10, [[-11, 1/4], [-10, 1/2], [-9, 1/4]], 1/2], [-11, [[-13, 1/4], [-11, 1/2], [-9, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-10, [[-11, 1/4], [-10, 1/2], [-9, 1/4]], 1/2], [-11, [[-13, 1/4], [-11, 1/2], [-9, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0028","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e92534af1843f4fc9a091bcb86dc531042afbc2732d13776ea6f462c5ecda1e1","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145028,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-10,1],[[-10,1],[-13,1],[-7,1]],[[1,2],[1,4],[1,4]]],[[-9,1],[[-10,1],[-8,1],[-9,1]],[[1,4],[1,4],[1,2]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-10, [-13, -10, -7], [[1, 4], [1, 2], [1, 4]]], [-9, [-10, -9, -8], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-10, [[-13, 1/4], [-10, 1/2], [-7, 1/4]], 9/2], [-9, [[-10, 1/4], [-9, 1/2], [-8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0029","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bf4347a47241bf115c0a9833671366b6363fedb2c3430f297f046ede8aa24778","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145029,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-8,1],[[-7,1],[-8,1],[-9,1]],[[1,4],[1,2],[1,4]]],[[-9,1],[[-13,1],[-5,1],[-9,1]],[[1,4],[1,4],[1,2]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-9, [-13, -9, -5], [[1, 4], [1, 2], [1, 4]]], [-8, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-8, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], 1/2], [-9, [[-13, 1/4], [-9, 1/2], [-5, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-8, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], 1/2], [-9, [[-13, 1/4], [-9, 1/2], [-5, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0030","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c065432ef2dc46a96fceb3715091ccb3a4f2ddd51bc247ecac2e692d7b2d1c2d","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145030,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-7,1],[[-6,1],[-7,1],[-8,1]],[[1,4],[1,2],[1,4]]],[[-8,1],[[-13,1],[-3,1],[-8,1]],[[1,4],[1,4],[1,2]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-8, [-13, -8, -3], [[1, 4], [1, 2], [1, 4]]], [-7, [-8, -7, -6], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2], [-8, [[-13, 1/4], [-8, 1/2], [-3, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2], [-8, [[-13, 1/4], [-8, 1/2], [-3, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0031","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7b2e346328d3d16597fc6e15d2289e7ba4dee15fe2d5a10b09eeb2b8c44792cf","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145031,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-6,1],[[-5,1],[-6,1],[-7,1]],[[1,4],[1,2],[1,4]]],[[-7,1],[[-6,1],[-7,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-7, [-8, -7, -6], [[1, 4], [1, 2], [1, 4]]], [-6, [-7, -6, -5], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-6, [[-7, 1/4], [-6, 1/2], [-5, 1/4]], 1/2], [-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-6, [[-7, 1/4], [-6, 1/2], [-5, 1/4]], 1/2], [-7, [[-8, 1/4], [-7, 1/2], [-6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0032","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b25b2493d853c045cce803491e21f1dcc41bab1c799bf9d824f7975154805c6a","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145032,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-5,1],[[-4,1],[-5,1],[-6,1]],[[1,4],[1,2],[1,4]]],[[-6,1],[[-4,1],[-6,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-6, [-8, -6, -4], [[1, 4], [1, 2], [1, 4]]], [-5, [-6, -5, -4], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-5, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], 1/2], [-6, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-5, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], 1/2], [-6, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0033","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bbe6781e08d43de6c17a9eb5a6ffb967fba0d4a84789ae760b93fd57cab22e78","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145033,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-4,1],[[-3,1],[-4,1],[-5,1]],[[1,4],[1,2],[1,4]]],[[-5,1],[[-2,1],[-5,1],[-8,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-5, [-8, -5, -2], [[1, 4], [1, 2], [1, 4]]], [-4, [-5, -4, -3], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-4, [[-5, 1/4], [-4, 1/2], [-3, 1/4]], 1/2], [-5, [[-8, 1/4], [-5, 1/2], [-2, 1/4]], 9/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-4, [[-5, 1/4], [-4, 1/2], [-3, 1/4]], 1/2], [-5, [[-8, 1/4], [-5, 1/2], [-2, 1/4]], 9/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0034","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9fea6597406a6003412673a2436a863536595bcc9c11ada6976d4537a4a92dbb","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145034,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-3,1],[[-2,1],[-3,1],[-4,1]],[[1,4],[1,2],[1,4]]],[[-4,1],[[-4,1],[-8,1],[0,1]],[[1,2],[1,4],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-4, [-8, -4, 0], [[1, 4], [1, 2], [1, 4]]], [-3, [-4, -3, -2], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-3, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1/2], [-4, [[-8, 1/4], [-4, 1/2], [0, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-3, [[-4, 1/4], [-3, 1/2], [-2, 1/4]], 1/2], [-4, [[-8, 1/4], [-4, 1/2], [0, 1/4]], 8]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0035","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"67f6706cb686599520247fa7c721a671dbb17967b172809ad4a830eb58138303","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145035,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-2,1],[[-1,1],[-2,1],[-3,1]],[[1,4],[1,2],[1,4]]],[[-3,1],[[-3,1],[-8,1],[2,1]],[[1,2],[1,4],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-3, [-8, -3, 2], [[1, 4], [1, 2], [1, 4]]], [-2, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2], [-3, [[-8, 1/4], [-3, 1/2], [2, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2], [-3, [[-8, 1/4], [-3, 1/2], [2, 1/4]], 25/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0036","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9c1a43f7f99143231fe81cec723ca183af7f024e7b94f02f1e15ebd30cabb795","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145036,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-1,1],[[-1,1],[-2,1],[0,1]],[[1,2],[1,4],[1,4]]],[[-2,1],[[-1,1],[-2,1],[-3,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-2, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]], [-1, [-2, -1, 0], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-1, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1/2], [-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-1, [[-2, 1/4], [-1, 1/2], [0, 1/4]], 1/2], [-2, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0037","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d5dd7d49c3d148d717ff5d301311c957bdf66557537350f36d10d849a022235d","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145037,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[-1,1],[[-1,1],[-3,1],[1,1]],[[1,2],[1,4],[1,4]]],[[0,1],[[-1,1],[0,1],[1,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[-1, [-3, -1, 1], [[1, 4], [1, 2], [1, 4]]], [0, [-1, 0, 1], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[-1, [[-3, 1/4], [-1, 1/2], [1, 1/4]], 2], [0, [[-1, 1/4], [0, 1/2], [1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[-1, [[-3, 1/4], [-1, 1/2], [1, 1/4]], 2], [0, [[-1, 1/4], [0, 1/2], [1, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0038","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"7f112754a721b63b37e21f0c2d9d0747b00d993954d81755f1ed3bb7d7332ad6","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145038,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[0,1],[[-3,1],[0,1],[3,1]],[[1,4],[1,2],[1,4]]],[[1,1],[[0,1],[1,1],[2,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[0, [-3, 0, 3], [[1, 4], [1, 2], [1, 4]]], [1, [0, 1, 2], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[0, [[-3, 1/4], [0, 1/2], [3, 1/4]], 9/2], [1, [[0, 1/4], [1, 1/2], [2, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[0, [[-3, 1/4], [0, 1/2], [3, 1/4]], 9/2], [1, [[0, 1/4], [1, 1/2], [2, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0039","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"b6d6f20c95a90de0527fd2391f3420c502cd581155ffd886737e4a3fd5c80981","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145039,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[1,1],[[-3,1],[1,1],[5,1]],[[1,4],[1,2],[1,4]]],[[2,1],[[1,1],[2,1],[3,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[1, [-3, 1, 5], [[1, 4], [1, 2], [1, 4]]], [2, [1, 2, 3], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[1, [[-3, 1/4], [1, 1/2], [5, 1/4]], 8], [2, [[1, 1/4], [2, 1/2], [3, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[1, [[-3, 1/4], [1, 1/2], [5, 1/4]], 8], [2, [[1, 1/4], [2, 1/2], [3, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0040","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"9357bb36351e5f34f46ae19191fee6a6adb683ca98614729f4d9a560bd6c5f44","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145040,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[2,1],[[-3,1],[2,1],[7,1]],[[1,4],[1,2],[1,4]]],[[3,1],[[2,1],[3,1],[4,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[2, [-3, 2, 7], [[1, 4], [1, 2], [1, 4]]], [3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[2, [[-3, 1/4], [2, 1/2], [7, 1/4]], 25/2], [3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[2, [[-3, 1/4], [2, 1/2], [7, 1/4]], 25/2], [3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0041","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"25a9758acc95a72bb2aed226553306f39975d152828733c119f0dcfa71820952","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145041,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[3,1],[[2,1],[3,1],[4,1]],[[1,4],[1,2],[1,4]]],[[4,1],[[3,1],[4,1],[5,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]], [4, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[3, [[2, 1/4], [3, 1/2], [4, 1/4]], 1/2], [4, [[3, 1/4], [4, 1/2], [5, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0042","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"41041b568a872a53567523e1a1c382618ea182fa378536db3390284302b6ab02","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145042,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[4,1],[[2,1],[4,1],[6,1]],[[1,4],[1,2],[1,4]]],[[5,1],[[4,1],[5,1],[6,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[4, [2, 4, 6], [[1, 4], [1, 2], [1, 4]]], [5, [4, 5, 6], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[4, [[2, 1/4], [4, 1/2], [6, 1/4]], 2], [5, [[4, 1/4], [5, 1/2], [6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[4, [[2, 1/4], [4, 1/2], [6, 1/4]], 2], [5, [[4, 1/4], [5, 1/2], [6, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0043","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e025679e2efe0addd5ff2dd01831f7fcf85dd7bacc4e6756c923bd41139d073e","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145043,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[5,1],[[2,1],[5,1],[8,1]],[[1,4],[1,2],[1,4]]],[[6,1],[[5,1],[6,1],[7,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[5, [2, 5, 8], [[1, 4], [1, 2], [1, 4]]], [6, [5, 6, 7], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[5, [[2, 1/4], [5, 1/2], [8, 1/4]], 9/2], [6, [[5, 1/4], [6, 1/2], [7, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[5, [[2, 1/4], [5, 1/2], [8, 1/4]], 9/2], [6, [[5, 1/4], [6, 1/2], [7, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0044","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0672f05efb0be5dcc6ea6a2696ad47f36f94f2876ac2062bb50dfac119d45a05","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145044,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[6,1],[[10,1],[2,1],[6,1]],[[1,4],[1,4],[1,2]]],[[7,1],[[6,1],[7,1],[8,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[6, [2, 6, 10], [[1, 4], [1, 2], [1, 4]]], [7, [6, 7, 8], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[6, [[2, 1/4], [6, 1/2], [10, 1/4]], 8], [7, [[6, 1/4], [7, 1/2], [8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[6, [[2, 1/4], [6, 1/2], [10, 1/4]], 8], [7, [[6, 1/4], [7, 1/2], [8, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0045","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fbe0911003f9fda24413e03d9bf6070ad43ec23cb94873a02c6b42ba948a715b","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145045,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[7,1],[[12,1],[2,1],[7,1]],[[1,4],[1,4],[1,2]]],[[8,1],[[7,1],[8,1],[9,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[7, [2, 7, 12], [[1, 4], [1, 2], [1, 4]]], [8, [7, 8, 9], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[7, [[2, 1/4], [7, 1/2], [12, 1/4]], 25/2], [8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[7, [[2, 1/4], [7, 1/2], [12, 1/4]], 25/2], [8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0046","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"636c284ac4c762b48d278dceabaae3ced4a2996774a0a63c7d1f9050d7ae15a1","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145046,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[8,1],[[7,1],[8,1],[9,1]],[[1,4],[1,2],[1,4]]],[[9,1],[[10,1],[8,1],[9,1]],[[1,4],[1,4],[1,2]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[8, [7, 8, 9], [[1, 4], [1, 2], [1, 4]]], [9, [8, 9, 10], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2], [9, [[8, 1/4], [9, 1/2], [10, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[8, [[7, 1/4], [8, 1/2], [9, 1/4]], 1/2], [9, [[8, 1/4], [9, 1/2], [10, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0047","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e71aedd434af7fbbbbd0a5d19e7654a97c2927a232fff953cd716bf49a170557","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145047,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[10,1],[[10,1],[11,1],[9,1]],[[1,2],[1,4],[1,4]]],[[9,1],[[11,1],[7,1],[9,1]],[[1,4],[1,4],[1,2]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[9, [7, 9, 11], [[1, 4], [1, 2], [1, 4]]], [10, [9, 10, 11], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[10, [[9, 1/4], [10, 1/2], [11, 1/4]], 1/2], [9, [[7, 1/4], [9, 1/2], [11, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[10, [[9, 1/4], [10, 1/2], [11, 1/4]], 1/2], [9, [[7, 1/4], [9, 1/2], [11, 1/4]], 2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0048","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"143b2cbe8eb3d22e48f3a38f5b3c1d69cffe24388252aead4de6f0f0ba98f321","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145048,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[10,1],[[10,1],[13,1],[7,1]],[[1,2],[1,4],[1,4]]],[[11,1],[[10,1],[11,1],[12,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[10, [7, 10, 13], [[1, 4], [1, 2], [1, 4]]], [11, [10, 11, 12], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[10, [[7, 1/4], [10, 1/2], [13, 1/4]], 9/2], [11, [[10, 1/4], [11, 1/2], [12, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[10, [[7, 1/4], [10, 1/2], [13, 1/4]], 9/2], [11, [[10, 1/4], [11, 1/2], [12, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0049","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"a03587060751f4323361517922e1349933425953e1f68a26d3b03233b9654115","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145049,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[11,1],[[11,1],[15,1],[7,1]],[[1,2],[1,4],[1,4]]],[[12,1],[[11,1],[12,1],[13,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[11, [7, 11, 15], [[1, 4], [1, 2], [1, 4]]], [12, [11, 12, 13], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[11, [[7, 1/4], [11, 1/2], [15, 1/4]], 8], [12, [[11, 1/4], [12, 1/2], [13, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[11, [[7, 1/4], [11, 1/2], [15, 1/4]], 8], [12, [[11, 1/4], [12, 1/2], [13, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0050","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b8f4aac1210d658de89414132bcbe889c3bc20be8498ab3e598b28d3be04c8d","family_id":"qf-mod45-mse-multi-theta-table","seed":202611145050,"semantic_template":"qf-mod45-mse-multi-theta-table/{\"laws\":[[[12,1],[[12,1],[17,1],[7,1]],[[1,2],[1,4],[1,4]]],[[13,1],[[12,1],[13,1],[14,1]],[[1,4],[1,2],[1,4]]]]}"},"fields":{"title":"Each Theta Has Its Own Complete Sampling Law and Risk","stem":"Use complete laws=[[12, [7, 12, 17], [[1, 4], [1, 2], [1, 4]]], [13, [12, 13, 14], [[1, 4], [1, 2], [1, 4]]]], each [theta,listed estimator values,aligned probabilities]. Theta labels are unique and every law is separately normalized. Give (law_risk_rows, certificate).","answer":{"kind":"expression","value":"([[12, [[7, 1/4], [12, 1/2], [17, 1/4]], 25/2], [13, [[12, 1/4], [13, 1/2], [14, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)","explanation":"For each law independently compute its sorted value/probability table and direct expected squared error. Do not pool theta values, average risks without a prior or silently reuse one sampling distribution for another theta. The exact result, retaining the requested order, is ([[12, [[7, 1/4], [12, 1/2], [17, 1/4]], 25/2], [13, [[12, 1/4], [13, 1/2], [14, 1/4]], 1/2]], MSE_is_computed_separately_under_each_complete_P_theta_law)."},"skills":["Each Theta Has Its Own Complete Sampling Law and Risk"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0051","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"401908f788aad32466e70b26dd1dee1e9d9b41aba255c58db9ff9578b0b1ad0d","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145051,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-11,1],[-12,1]],\"theta\":[-12,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-12, complete listed estimator values=[-12, -11] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-12, [[-12, 1, 0, 0], [-11, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0052","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"687811e2d4d3dc157074c696d049a984fdfe0115660f0e5cea35d611428880a3","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145052,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-11,1],[-9,1]],\"theta\":[-11,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-11, complete listed estimator values=[-11, -9] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-11, [[-11, 1, 0, 0], [-9, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-11, [[-11, 1, 0, 0], [-9, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0053","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91efcddaaf1df7888eb43d7f6a330aaeaa7a3c7cd3a93a6cbbbbf77849b755c1","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145053,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-10,1],[-7,1]],\"theta\":[-10,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-10, complete listed estimator values=[-10, -7] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-10, [[-10, 1, 0, 0], [-7, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-10, [[-10, 1, 0, 0], [-7, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0054","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"233c53a9828f2626ca56576c980166ef8dcb96ef4f92607c3e58d7fd23308dde","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145054,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-5,1],[-9,1]],\"theta\":[-9,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-9, complete listed estimator values=[-9, -5] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-9, [[-9, 1, 0, 0], [-5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-9, [[-9, 1, 0, 0], [-5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0055","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"203c5edaab8b6919e058f531bf5ed043ab43b6ed5cfbf9af4313fdfcaa6b041d","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145055,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-3,1],[-8,1]],\"theta\":[-8,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-8, complete listed estimator values=[-8, -3] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-8, [[-8, 1, 0, 0], [-3, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-8, [[-8, 1, 0, 0], [-3, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0056","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d74b860cebdcdafed75435137d1d2c9c5219fa6a3aeec06bc5b60427ae6b99ff","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145056,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-6,1],[-7,1]],\"theta\":[-7,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-7, complete listed estimator values=[-7, -6] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-7, [[-7, 1, 0, 0], [-6, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0057","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6af9a9e6dc7ab8d1c9c9ab111494b1320cc033ec610c3b39b0ee0638b41e09ad","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145057,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-4,1],[-6,1]],\"theta\":[-6,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-6, complete listed estimator values=[-6, -4] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-6, [[-6, 1, 0, 0], [-4, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-6, [[-6, 1, 0, 0], [-4, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0058","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"49b79c2678f691ce781e58b186397fe0268523e847088174288eda5eb1d38380","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145058,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-2,1],[-5,1]],\"theta\":[-5,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-5, complete listed estimator values=[-5, -2] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-5, [[-5, 1, 0, 0], [-2, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-5, [[-5, 1, 0, 0], [-2, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0059","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"833d4cc7762733e38817c674bc4aebe23d537f46a6d6bdfc8799a27771c765a5","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145059,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-4,1],[0,1]],\"theta\":[-4,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-4, complete listed estimator values=[-4, 0] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-4, [[-4, 1, 0, 0], [0, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-4, [[-4, 1, 0, 0], [0, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0060","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"054df7cba179abab291db2421f4d572e5c1613e9f9a3e0ce903bc0e13dc04787","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145060,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-3,1],[2,1]],\"theta\":[-3,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-3, complete listed estimator values=[-3, 2] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-3, [[-3, 1, 0, 0], [2, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-3, [[-3, 1, 0, 0], [2, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0061","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6e59babb04bbb43a4ecb721cc4c52dd71b039dd8bde77c3a194e327b7468384c","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145061,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[-1,1],[-2,1]],\"theta\":[-2,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-2, complete listed estimator values=[-2, -1] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-2, [[-2, 1, 0, 0], [-1, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-2, [[-2, 1, 0, 0], [-1, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0062","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ee3acb7c1cc983596de24ad2108920c97a9f3242894a8e6d92890a4594578deb","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145062,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[-1,1],[1,1]],\"theta\":[-1,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=-1, complete listed estimator values=[-1, 1] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(-1, [[-1, 1, 0, 0], [1, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (-1, [[-1, 1, 0, 0], [1, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0063","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"de7e93ae0affe55fbc50a517fb8a46288ca815b3223d46d6b703a0566cdb7451","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145063,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[0,1],[3,1]],\"theta\":[0,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=0, complete listed estimator values=[0, 3] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(0, [[0, 1, 0, 0], [3, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (0, [[0, 1, 0, 0], [3, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0064","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d7503ff65f9f3a14f19aaeb5e030aea7ab15a8926fa9563d4d95a3b9ceafece2","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145064,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[1,1],[5,1]],\"theta\":[1,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=1, complete listed estimator values=[1, 5] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(1, [[1, 1, 0, 0], [5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (1, [[1, 1, 0, 0], [5, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0065","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6481ce08ee444466d42482209ae6f80ad823fd67646bfe4e147825d11d19cbca","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145065,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[2,1],[7,1]],\"theta\":[2,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=2, complete listed estimator values=[2, 7] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(2, [[2, 1, 0, 0], [7, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (2, [[2, 1, 0, 0], [7, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0066","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fa911bf58e33ce778551758c451e52fab9d0f036b420d2eedf2f2010c6c926a8","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145066,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[3,1],[4,1]],\"theta\":[3,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=3, complete listed estimator values=[3, 4] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(3, [[3, 1, 0, 0], [4, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (3, [[3, 1, 0, 0], [4, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0067","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3ea4518fdf9876076a1ad6881b8d849f7a0f48958aaeea9daefda65e6727f064","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145067,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[4,1],[6,1]],\"theta\":[4,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=4, complete listed estimator values=[4, 6] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(4, [[4, 1, 0, 0], [6, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (4, [[4, 1, 0, 0], [6, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0068","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"c8786178878ae4a8eb136412670fdcfa82746e2dbe6ce3d66d9e1d78e77a788d","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145068,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[5,1],[8,1]],\"theta\":[5,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=5, complete listed estimator values=[5, 8] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(5, [[5, 1, 0, 0], [8, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (5, [[5, 1, 0, 0], [8, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0069","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5ce06feaab6de457c1899ed369c973f9f8d1d5cd50f1aa5e93f0b875d11ffb03","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145069,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[10,1],[6,1]],\"theta\":[6,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=6, complete listed estimator values=[6, 10] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(6, [[6, 1, 0, 0], [10, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (6, [[6, 1, 0, 0], [10, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0070","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"8b8356e9410e5580efca2fe92707d5334572028f84726f327424689193e2355b","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145070,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[12,1],[7,1]],\"theta\":[7,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=7, complete listed estimator values=[7, 12] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(7, [[7, 1, 0, 0], [12, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (7, [[7, 1, 0, 0], [12, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0071","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f3adaad0c8726ac9b2c8ee1a61139c8ae0151eacb1510decd8185974b57f1296","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145071,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[8,1],[9,1]],\"theta\":[8,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=8, complete listed estimator values=[8, 9] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(8, [[8, 1, 0, 0], [9, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (8, [[8, 1, 0, 0], [9, 0, 1, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0072","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"4a3f004730bdc899bda14ea972f35f91cd8cfc44fdd34b7010e4d6c02d790470","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145072,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[0,1],[1,1]],\"support\":[[11,1],[9,1]],\"theta\":[9,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=9, complete listed estimator values=[9, 11] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(9, [[9, 1, 0, 0], [11, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (9, [[9, 1, 0, 0], [11, 0, 4, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0073","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e646361ddb0d840dd112e24f8b13fc537d57c503005c366f4e9f5f586a70a19b","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145073,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[10,1],[13,1]],\"theta\":[10,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=10, complete listed estimator values=[10, 13] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(10, [[10, 1, 0, 0], [13, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (10, [[10, 1, 0, 0], [13, 0, 9, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0074","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"788519aeafecba87604573e34de98f67747e91a709d7ba3922c498d10908e31d","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145074,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[11,1],[15,1]],\"theta\":[11,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=11, complete listed estimator values=[11, 15] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(11, [[11, 1, 0, 0], [15, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (11, [[11, 1, 0, 0], [15, 0, 16, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0075","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"40641a1edde648e3d0fd492064bc82a66eb6358c92bd5eadf7d8b65d95c4f841","family_id":"qf-mod45-mse-zero-iff-almost-sure","seed":202611145075,"semantic_template":"qf-mod45-mse-zero-iff-almost-sure/{\"probabilities\":[[1,1],[0,1]],\"support\":[[12,1],[17,1]],\"theta\":[12,1]}"},"fields":{"title":"Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row","stem":"Use scalar theta=12, complete listed estimator values=[12, 17] and aligned probabilities=[1, 0] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Give (theta, weighted_error_rows, MSE, zero_MSE, almost_surely_equal_theta, equivalence_holds, certificate).","answer":{"kind":"expression","value":"(12, [[12, 1, 0, 0], [17, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)","explanation":"Keep every squared/weighted loss. Nonnegative weighted squares imply MSE0 exactly when every positive-mass value equals theta. Zero-mass listed values may differ. The final equivalence flag may be Yes even when zero_MSE and almost_surely_equal_theta are both No. The exact result, retaining the requested order, is (12, [[12, 1, 0, 0], [17, 0, 25, 0]], 0, Yes, Yes, Yes, nonnegative_squared_terms_make_zero_MSE_equivalent_to_T_equals_theta_almost_surely)."},"skills":["Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0076","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"ce981bdb429ca673ebd5c24cd52d6868981cde2adcae41dee12d64b1f8bc9d0b","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145076,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-11,1],[-12,1],[-13,1]],\"theta\":[-12,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-12, complete listed estimator values=[-13, -12, -11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-12, -13, 1, -12, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-12, -13, 1, -12, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0077","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"5b49b5d7acca79d01250ebabb2fe7cc268fd2c790e010fe5cdecbe45acdd3247","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145077,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-11,1],[-13,1],[-9,1]],\"theta\":[-11,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-11, complete listed estimator values=[-13, -11, -9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-11, -13, 4, -11, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-11, -13, 4, -11, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0078","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2bf25abea5b7ff89d1f796bdac67ea23fad336bf7f44fc467296c7d956d0cb5c","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145078,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-10,1],[-13,1],[-7,1]],\"theta\":[-10,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-10, complete listed estimator values=[-13, -10, -7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-10, -13, 9, -10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-10, -13, 9, -10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0079","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"54a9a3cbafba113b0502d8c86043642565bc6887b3b5b256642dce2b0e9f7621","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145079,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-5,1],[-9,1]],\"theta\":[-9,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-9, complete listed estimator values=[-13, -9, -5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-9, -13, 16, -9, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-9, -13, 16, -9, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0080","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"2578915ea3460ac5856ae5037fa1280eb419cd99a6399dd3e2b44c0a62639a8b","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145080,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-13,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[-13,1],[-3,1],[-8,1]],\"theta\":[-8,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-8, complete listed estimator values=[-13, -8, -3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-8, -13, 25, -8, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-8, -13, 25, -8, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0081","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"dfbac74d9a1e83d9a3878f9fa7bc6242b2d5d1b3f3ae762ddebc41eb8f99db04","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145081,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-6,1],[-7,1],[-8,1]],\"theta\":[-7,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-7, complete listed estimator values=[-8, -7, -6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-7, -8, 1, -7, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-7, -8, 1, -7, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0082","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"d99d58f81fa1ef3efe20d71b9f921f426680debcc553d5e87fbba4b364fa7123","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145082,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-4,1],[-6,1],[-8,1]],\"theta\":[-6,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-6, complete listed estimator values=[-8, -6, -4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-6, -8, 4, -6, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-6, -8, 4, -6, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0083","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"91200443ff3717718522df11e692246fb3aa03fde7a623794b52acd209261765","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145083,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-2,1],[-5,1],[-8,1]],\"theta\":[-5,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-5, complete listed estimator values=[-8, -5, -2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-5, -8, 9, -5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-5, -8, 9, -5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0084","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"f7dd87226a97ebc9e729582b6a18e34f821d0c4fc122edea5e74f873656a1f3a","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145084,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-4,1],[-8,1],[0,1]],\"theta\":[-4,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-4, complete listed estimator values=[-8, -4, 0] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-4, -8, 16, -4, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-4, -8, 16, -4, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0085","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"bb85ca5c5381948121fff459c1bc67ae0cc46fdbe6171d1242f24e4798794f18","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145085,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-8,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-3,1],[-8,1],[2,1]],\"theta\":[-3,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-3, complete listed estimator values=[-8, -3, 2] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-8. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-3, -8, 25, -3, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-3, -8, 25, -3, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0086","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e5ec9b285b64dcd8f1f62a6582437207725ac943d13297508e3a197af0dfd18d","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145086,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-1,1],[-2,1],[-3,1]],\"theta\":[-2,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-2, complete listed estimator values=[-3, -2, -1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-2, -3, 1, -2, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-2, -3, 1, -2, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0087","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"fac5bea2ce444b42265c51738b354d00aa3beabef87a08d8ed7d03fefc38929a","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145087,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[-1,1],[-3,1],[1,1]],\"theta\":[-1,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=-1, complete listed estimator values=[-3, -1, 1] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(-1, -3, 4, -1, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (-1, -3, 4, -1, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0088","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e704ab11cd468315bc1d161102ef4bd3b63ca2addce748b27db50b4a33582d67","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145088,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[0,1],[3,1]],\"theta\":[0,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=0, complete listed estimator values=[-3, 0, 3] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(0, -3, 9, 0, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (0, -3, 9, 0, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0089","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6091680ea5909ec701353cc1b7c22e153afdc45f5a5747773c423c4909a4b127","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145089,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[1,1],[5,1]],\"theta\":[1,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=1, complete listed estimator values=[-3, 1, 5] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(1, -3, 16, 1, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (1, -3, 16, 1, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0090","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e79ad904d81f5c799635ace2480c09449124c630ca4415b25b46dad55d8916f3","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145090,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[-3,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[-3,1],[2,1],[7,1]],\"theta\":[2,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=2, complete listed estimator values=[-3, 2, 7] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(2, -3, 25, 2, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (2, -3, 25, 2, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0091","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"cbccd4c2771946a344ff7c8e8b66c1a0fb9f953329c8d839ca3614491c5d2935","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145091,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[3,1],[4,1]],\"theta\":[3,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=3, complete listed estimator values=[2, 3, 4] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(3, 2, 1, 3, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (3, 2, 1, 3, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0092","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"0ba79fbd4dd52ebfb92d50dcb5ef297eb875274919d78b3f3808d44b4102a987","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145092,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[4,1],[6,1]],\"theta\":[4,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=4, complete listed estimator values=[2, 4, 6] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(4, 2, 4, 4, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (4, 2, 4, 4, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0093","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"e57eda3af41752169b8a884d052f6efc75983c455f54646fe0df82b3573bbcb3","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145093,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[2,1],[5,1],[8,1]],\"theta\":[5,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=5, complete listed estimator values=[2, 5, 8] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(5, 2, 9, 5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (5, 2, 9, 5, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0094","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"05089fd65e2c1cf59b00b740e5741f2b02e89d1d3c3427699a9c81f97fa8d8c7","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145094,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[10,1],[2,1],[6,1]],\"theta\":[6,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=6, complete listed estimator values=[2, 6, 10] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(6, 2, 16, 6, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (6, 2, 16, 6, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0095","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"3f50fba6c59e669c05495c405ce455ede50557609932c9cb7aef0dc7f48da3f2","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145095,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[2,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[12,1],[2,1],[7,1]],\"theta\":[7,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=7, complete listed estimator values=[2, 7, 12] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(7, 2, 25, 7, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (7, 2, 25, 7, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0096","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"689b92f39ddeba843beaf700bff0665a91fe5da49edc55332412e0c48edd992a","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145096,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,4],[1,2],[1,4]],\"support\":[[7,1],[8,1],[9,1]],\"theta\":[8,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=8, complete listed estimator values=[7, 8, 9] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(8, 7, 1, 8, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (8, 7, 1, 8, Yes, 1/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0097","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"091c9be413f3e8bb2e113bd675e8fa34d9ad3f327c118432d71098092cdcfadc","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145097,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,4],[1,4],[1,2]],\"support\":[[11,1],[7,1],[9,1]],\"theta\":[9,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=9, complete listed estimator values=[7, 9, 11] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(9, 7, 4, 9, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (9, 7, 4, 9, Yes, 2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0098","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6b4fed3adbf11e6bb4cf6c3f4b3ffac0ea1dbf1e2d6e0586ee3b9ca07028c73f","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145098,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[10,1],[13,1],[7,1]],\"theta\":[10,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=10, complete listed estimator values=[7, 10, 13] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(10, 7, 9, 10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (10, 7, 9, 10, Yes, 9/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0099","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"6fcfa2edac900a0797696df78f4adb896ba9f5435a3fdcfcc79df148f73bdd20","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145099,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[11,1],[15,1],[7,1]],\"theta\":[11,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=11, complete listed estimator values=[7, 11, 15] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(11, 7, 16, 11, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (11, 7, 16, 11, Yes, 8, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}
{"schema_version":"ocme-question-localization-v0.1","question_id":"q-20261015-advance-20261114-r05-estimator-mean-squared-error-0100","batch_id":"batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations","locale":"en","translation_state":"ai_candidate","source":{"question_sha256":"92f3d145a490aba51311843130c2756c680747174dedd875369eb34f6312535a","family_id":"qf-mod45-mse-observed-unbiased-boundary","seed":202611145100,"semantic_template":"qf-mod45-mse-observed-unbiased-boundary/{\"observed\":[7,1],\"probabilities\":[[1,2],[1,4],[1,4]],\"support\":[[12,1],[17,1],[7,1]],\"theta\":[12,1]}"},"fields":{"title":"One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency","stem":"Use scalar theta=12, complete listed estimator values=[7, 12, 17] and aligned probabilities=[1/4, 1/2, 1/4] under this same P_theta. Values and probabilities are exact rationals; probabilities are nonnegative and sum to1. Listed zero-mass rows remain in the data and contribute0. Output rows sort values numerically without separating them from their probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, mean, unbiased, MSE, observed_error_equals_MSE, unbiased_with_positive_MSE, observed_or_unbiased_consistency_claim, boundary).","answer":{"kind":"expression","value":"(12, 7, 25, 12, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)","explanation":"Compute this squared error, the law mean/unbiasedness and the complete MSE separately. observed_error_equals_MSE may accidentally be Yes; the fixed No does not negate that equality. Unbiasedness can coexist with positive MSE. No n-indexed consistency or asymptotic-optimality certificate is supplied by this finite law. The exact result, retaining the requested order, is (12, 7, 25, 12, Yes, 25/2, No, Yes, No, one_observed_squared_error_is_not_MSE_and_unbiased_does_not_imply_zero_MSE_or_consistency)."},"skills":["One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency"],"verification_warning":"This is a candidate exercise. Mechanical recomputation checks only the stated conditions and this instance; it is neither a formal proof nor human review."}}