# Exact Theta-Indexed Expected Loss, Almost-Sure Equality and Observed-Error Limits

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batch-20261015-advance-20261114-r05-estimator-mean-squared-error-foundations · 2026-10-15 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.

Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.

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## 1. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0001/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 2. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0002/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 3. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0003/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 4. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0004/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 5. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0005/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 6. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0006/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 7. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0007/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 8. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0008/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 9. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0009/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 10. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0010/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 11. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0011/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 12. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0012/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 13. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0013/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 14. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0014/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 15. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0015/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 16. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0016/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 17. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0017/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 18. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0018/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 19. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0019/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 20. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0020/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 21. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0021/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 22. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0022/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 23. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0023/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 24. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0024/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 25. MSE Is the Complete Expected Squared Error under One Theta Law

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).

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0025/)

Mean squared error of an estimator · qf-mod45-mse-single-theta-exact · candidate





## 26. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0026/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 27. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0027/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 28. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0028/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 29. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0029/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 30. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0030/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 31. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0031/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 32. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0032/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 33. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0033/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 34. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0034/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 35. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0035/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 36. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0036/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 37. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0037/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 38. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0038/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 39. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0039/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 40. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0040/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 41. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0041/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 42. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0042/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 43. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0043/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 44. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0044/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 45. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0045/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 46. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0046/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 47. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0047/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 48. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0048/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 49. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0049/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 50. Each Theta Has Its Own Complete Sampling Law and Risk

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).

View answer and explanation([[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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0050/)

Mean squared error of an estimator · qf-mod45-mse-multi-theta-table · candidate





## 51. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0051/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 52. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0052/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 53. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0053/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 54. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0054/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 55. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0055/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 56. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0056/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 57. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0057/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 58. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0058/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 59. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0059/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 60. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0060/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 61. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0061/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 62. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0062/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 63. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0063/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 64. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0064/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 65. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0065/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 66. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0066/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 67. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0067/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 68. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0068/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 69. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0069/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 70. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0070/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 71. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0071/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 72. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0072/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 73. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0073/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 74. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0074/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 75. Zero MSE Means Equality to Theta Almost Surely, Not at Every Zero-Mass Row

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).

Almost surely constrains positive-mass values only. Zero-mass rows remain and need not equal theta. equivalence_holds is separate from zero_MSE.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0075/)

Mean squared error of an estimator · qf-mod45-mse-zero-iff-almost-sure · candidate





## 76. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0076/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 77. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0077/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 78. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0078/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 79. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0079/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 80. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0080/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 81. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0081/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 82. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0082/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 83. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0083/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 84. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0084/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 85. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0085/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 86. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0086/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 87. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(-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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0087/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 88. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0088/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 89. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0089/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 90. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0090/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 91. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0091/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 92. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0092/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 93. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0093/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 94. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0094/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 95. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0095/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 96. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0096/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 97. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0097/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 98. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0098/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 99. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0099/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate





## 100. One Observed Loss and Unbiasedness Do Not Determine MSE or Consistency

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).

observed_error_equals_MSE may accidentally be Yes. The final No does not negate that equality or turn unbiasedness into zero MSE or consistency.

View answer and explanation(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)

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).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261015-advance-20261114-r05-estimator-mean-squared-error-0100/)

Mean squared error of an estimator · qf-mod45-mse-observed-unbiased-boundary · candidate



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