Theta-Specific Bias-Variance Identities and Centering
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batch-20261015-advance-20261115-r03-bias-variance-decomposition-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 Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-12, complete estimator values=[-12, -11, -10] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-12, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], -11, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-12, [[-12, 1/4], [-11, 1/2], [-10, 1/4]], -11, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
2. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-11, complete estimator values=[-11, -9, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-11, [[-11, 1/4], [-9, 1/2], [-7, 1/4]], -9, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-11, [[-11, 1/4], [-9, 1/2], [-7, 1/4]], -9, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
3. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-10, complete estimator values=[-10, -7, -4] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-10, [[-10, 1/4], [-7, 1/2], [-4, 1/4]], -7, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-10, [[-10, 1/4], [-7, 1/2], [-4, 1/4]], -7, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
4. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-9, complete estimator values=[-9, -8, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-9, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], -8, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-9, [[-9, 1/4], [-8, 1/2], [-7, 1/4]], -8, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
5. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-8, complete estimator values=[-8, -6, -4] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-8, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], -6, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-8, [[-8, 1/4], [-6, 1/2], [-4, 1/4]], -6, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
6. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-7, complete estimator values=[-7, -4, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-7, [[-7, 1/4], [-4, 1/2], [-1, 1/4]], -4, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-7, [[-7, 1/4], [-4, 1/2], [-1, 1/4]], -4, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
7. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-6, complete estimator values=[-6, -5, -4] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-6, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], -5, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-6, [[-6, 1/4], [-5, 1/2], [-4, 1/4]], -5, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
8. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-5, complete estimator values=[-5, -3, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-5, [[-5, 1/4], [-3, 1/2], [-1, 1/4]], -3, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-5, [[-5, 1/4], [-3, 1/2], [-1, 1/4]], -3, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
9. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-4, complete estimator values=[-4, -1, 2] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-4, [[-4, 1/4], [-1, 1/2], [2, 1/4]], -1, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-4, [[-4, 1/4], [-1, 1/2], [2, 1/4]], -1, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
10. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-3, complete estimator values=[-3, -2, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-3, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], -2, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-3, [[-3, 1/4], [-2, 1/2], [-1, 1/4]], -2, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
11. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-2, complete estimator values=[-2, 0, 2] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-2, [[-2, 1/4], [0, 1/2], [2, 1/4]], 0, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-2, [[-2, 1/4], [0, 1/2], [2, 1/4]], 0, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
12. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=-1, complete estimator values=[-1, 2, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(-1, [[-1, 1/4], [2, 1/2], [5, 1/4]], 2, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (-1, [[-1, 1/4], [2, 1/2], [5, 1/4]], 2, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
13. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=0, complete estimator values=[0, 1, 2] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(0, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (0, [[0, 1/4], [1, 1/2], [2, 1/4]], 1, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
14. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=1, complete estimator values=[1, 3, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(1, [[1, 1/4], [3, 1/2], [5, 1/4]], 3, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (1, [[1, 1/4], [3, 1/2], [5, 1/4]], 3, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
15. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=2, complete estimator values=[2, 5, 8] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(2, [[2, 1/4], [5, 1/2], [8, 1/4]], 5, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (2, [[2, 1/4], [5, 1/2], [8, 1/4]], 5, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
16. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=3, complete estimator values=[3, 4, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(3, [[3, 1/4], [4, 1/2], [5, 1/4]], 4, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (3, [[3, 1/4], [4, 1/2], [5, 1/4]], 4, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
17. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=4, complete estimator values=[4, 6, 8] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(4, [[4, 1/4], [6, 1/2], [8, 1/4]], 6, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (4, [[4, 1/4], [6, 1/2], [8, 1/4]], 6, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
18. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=5, complete estimator values=[5, 8, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(5, [[5, 1/4], [8, 1/2], [11, 1/4]], 8, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (5, [[5, 1/4], [8, 1/2], [11, 1/4]], 8, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
19. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=6, complete estimator values=[6, 7, 8] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(6, [[6, 1/4], [7, 1/2], [8, 1/4]], 7, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (6, [[6, 1/4], [7, 1/2], [8, 1/4]], 7, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
20. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=7, complete estimator values=[7, 9, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(7, [[7, 1/4], [9, 1/2], [11, 1/4]], 9, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (7, [[7, 1/4], [9, 1/2], [11, 1/4]], 9, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
21. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=8, complete estimator values=[8, 11, 14] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(8, [[8, 1/4], [11, 1/2], [14, 1/4]], 11, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (8, [[8, 1/4], [11, 1/2], [14, 1/4]], 11, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
22. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=9, complete estimator values=[9, 10, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(9, [[9, 1/4], [10, 1/2], [11, 1/4]], 10, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (9, [[9, 1/4], [10, 1/2], [11, 1/4]], 10, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
23. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=10, complete estimator values=[10, 12, 14] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(10, [[10, 1/4], [12, 1/2], [14, 1/4]], 12, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (10, [[10, 1/4], [12, 1/2], [14, 1/4]], 12, 2, 2, 6, 6, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
24. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=11, complete estimator values=[11, 14, 17] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(11, [[11, 1/4], [14, 1/2], [17, 1/4]], 14, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (11, [[11, 1/4], [14, 1/2], [17, 1/4]], 14, 3, 9/2, 27/2, 27/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
25. MSE Equals Variance plus Squared Bias under One Complete Theta Law
Use scalar theta=12, complete estimator values=[12, 13, 14] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (theta, law_rows, mean, bias, variance, MSE, variance_plus_bias_squared, identity_holds, certificate).
View answer and explanation
(12, [[12, 1/4], [13, 1/2], [14, 1/4]], 13, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law)
Compute mean, bias, variance and MSE separately under the same law, then compare the direct MSE with variance+bias^2. The identity is a finite scalar-law certificate, not consistency or an asymptotic claim. The exact result, retaining the requested order, is (12, [[12, 1/4], [13, 1/2], [14, 1/4]], 13, 1, 1/2, 3/2, 3/2, Yes, MSE_equals_variance_plus_squared_bias_under_one_complete_sampling_law).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-single-theta-identity · candidate
26. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-12, complete estimator values=[-11, -10, -9] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-10, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-10, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
27. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-11, complete estimator values=[-10, -9, -8] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-9, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-9, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
28. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-10, complete estimator values=[-9, -8, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-8, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-8, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
29. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-9, complete estimator values=[-8, -7, -6] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-7, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-7, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
30. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-8, complete estimator values=[-7, -6, -5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-6, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-6, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
31. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-7, complete estimator values=[-6, -5, -4] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-5, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-5, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
32. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-6, complete estimator values=[-5, -4, -3] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-4, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-4, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
33. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-5, complete estimator values=[-4, -3, -2] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-3, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-3, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
34. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-4, complete estimator values=[-3, -2, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-2, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-2, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
35. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-3, complete estimator values=[-2, -1, 0] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(-1, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (-1, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
36. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-2, complete estimator values=[-1, 0, 1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(0, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (0, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
37. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=-1, complete estimator values=[0, 1, 2] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(1, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (1, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
38. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=0, complete estimator values=[1, 2, 3] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(2, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (2, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
39. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=1, complete estimator values=[2, 3, 4] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(3, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (3, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
40. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=2, complete estimator values=[3, 4, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(4, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (4, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
41. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=3, complete estimator values=[4, 5, 6] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(5, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (5, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
42. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=4, complete estimator values=[5, 6, 7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(6, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (6, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
43. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=5, complete estimator values=[6, 7, 8] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(7, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (7, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
44. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=6, complete estimator values=[7, 8, 9] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(8, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (8, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
45. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=7, complete estimator values=[8, 9, 10] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(9, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (9, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
46. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=8, complete estimator values=[9, 10, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(10, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (10, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
47. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=9, complete estimator values=[10, 11, 12] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(11, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (11, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
48. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=10, complete estimator values=[11, 12, 13] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(12, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (12, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
49. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=11, complete estimator values=[12, 13, 14] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(13, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (13, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
50. The Cross Term Vanishes Because the Centered Estimator Has Mean Zero
Use scalar theta=12, complete estimator values=[13, 14, 15] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Give (mean, centered_mean, bias, cross_term, cross_term_zero, certificate).
View answer and explanation
(14, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes)
Compute E[T−E[T]], bias and twice their product. The cross term vanishes by the centering identity, not because every observed deviation or the bias itself must be zero. The exact result, retaining the requested order, is (14, 0, 2, 0, Yes, E_of_T_minus_E_T_is_zero_so_the_cross_term_vanishes).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-cross-term-cancellation · candidate
51. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-12, [-12, -11, -10], [[1, 4], [1, 2], [1, 4]]], [-11, [-12, -11, -10], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-11, -11, 0, 1/2, 1/2, 1/2, Yes], [-12, -11, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-11, -11, 0, 1/2, 1/2, 1/2, Yes], [-12, -11, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
52. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-11, [-11, -9, -7], [[1, 4], [1, 2], [1, 4]]], [-10, [-11, -10, -9], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-10, -10, 0, 1/2, 1/2, 1/2, Yes], [-11, -9, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-10, -10, 0, 1/2, 1/2, 1/2, Yes], [-11, -9, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
53. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-10, [-10, -7, -4], [[1, 4], [1, 2], [1, 4]]], [-9, [-10, -9, -8], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-10, -7, 3, 9/2, 27/2, 27/2, Yes], [-9, -9, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-10, -7, 3, 9/2, 27/2, 27/2, Yes], [-9, -9, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
54. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-9, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]], [-8, [-9, -8, -7], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-8, -8, 0, 1/2, 1/2, 1/2, Yes], [-9, -8, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-8, -8, 0, 1/2, 1/2, 1/2, Yes], [-9, -8, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
55. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-8, [-8, -6, -4], [[1, 4], [1, 2], [1, 4]]], [-7, [-8, -7, -6], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-7, -7, 0, 1/2, 1/2, 1/2, Yes], [-8, -6, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-7, -7, 0, 1/2, 1/2, 1/2, Yes], [-8, -6, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
56. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-7, [-7, -4, -1], [[1, 4], [1, 2], [1, 4]]], [-6, [-7, -6, -5], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-6, -6, 0, 1/2, 1/2, 1/2, Yes], [-7, -4, 3, 9/2, 27/2, 27/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-6, -6, 0, 1/2, 1/2, 1/2, Yes], [-7, -4, 3, 9/2, 27/2, 27/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
57. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-6, [-6, -5, -4], [[1, 4], [1, 2], [1, 4]]], [-5, [-6, -5, -4], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-5, -5, 0, 1/2, 1/2, 1/2, Yes], [-6, -5, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-5, -5, 0, 1/2, 1/2, 1/2, Yes], [-6, -5, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
58. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-5, [-5, -3, -1], [[1, 4], [1, 2], [1, 4]]], [-4, [-5, -4, -3], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-4, -4, 0, 1/2, 1/2, 1/2, Yes], [-5, -3, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-4, -4, 0, 1/2, 1/2, 1/2, Yes], [-5, -3, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
59. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-4, [-4, -1, 2], [[1, 4], [1, 2], [1, 4]]], [-3, [-4, -3, -2], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-3, -3, 0, 1/2, 1/2, 1/2, Yes], [-4, -1, 3, 9/2, 27/2, 27/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-3, -3, 0, 1/2, 1/2, 1/2, Yes], [-4, -1, 3, 9/2, 27/2, 27/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
60. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-3, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]], [-2, [-3, -2, -1], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-2, -2, 0, 1/2, 1/2, 1/2, Yes], [-3, -2, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-2, -2, 0, 1/2, 1/2, 1/2, Yes], [-3, -2, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
61. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-2, [-2, 0, 2], [[1, 4], [1, 2], [1, 4]]], [-1, [-2, -1, 0], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-1, -1, 0, 1/2, 1/2, 1/2, Yes], [-2, 0, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-1, -1, 0, 1/2, 1/2, 1/2, Yes], [-2, 0, 2, 2, 6, 6, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
62. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[-1, [-1, 2, 5], [[1, 4], [1, 2], [1, 4]]], [0, [-1, 0, 1], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[-1, 2, 3, 9/2, 27/2, 27/2, Yes], [0, 0, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[-1, 2, 3, 9/2, 27/2, 27/2, Yes], [0, 0, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
63. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[0, [0, 1, 2], [[1, 4], [1, 2], [1, 4]]], [1, [0, 1, 2], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[0, 1, 1, 1/2, 3/2, 3/2, Yes], [1, 1, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[0, 1, 1, 1/2, 3/2, 3/2, Yes], [1, 1, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
64. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[1, [1, 3, 5], [[1, 4], [1, 2], [1, 4]]], [2, [1, 2, 3], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[1, 3, 2, 2, 6, 6, Yes], [2, 2, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[1, 3, 2, 2, 6, 6, Yes], [2, 2, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
65. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[2, [2, 5, 8], [[1, 4], [1, 2], [1, 4]]], [3, [2, 3, 4], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[2, 5, 3, 9/2, 27/2, 27/2, Yes], [3, 3, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[2, 5, 3, 9/2, 27/2, 27/2, Yes], [3, 3, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
66. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[3, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]], [4, [3, 4, 5], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[3, 4, 1, 1/2, 3/2, 3/2, Yes], [4, 4, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[3, 4, 1, 1/2, 3/2, 3/2, Yes], [4, 4, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
67. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[4, [4, 6, 8], [[1, 4], [1, 2], [1, 4]]], [5, [4, 5, 6], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[4, 6, 2, 2, 6, 6, Yes], [5, 5, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[4, 6, 2, 2, 6, 6, Yes], [5, 5, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
68. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[5, [5, 8, 11], [[1, 4], [1, 2], [1, 4]]], [6, [5, 6, 7], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[5, 8, 3, 9/2, 27/2, 27/2, Yes], [6, 6, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[5, 8, 3, 9/2, 27/2, 27/2, Yes], [6, 6, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
69. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[6, [6, 7, 8], [[1, 4], [1, 2], [1, 4]]], [7, [6, 7, 8], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[6, 7, 1, 1/2, 3/2, 3/2, Yes], [7, 7, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[6, 7, 1, 1/2, 3/2, 3/2, Yes], [7, 7, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
70. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[7, [7, 9, 11], [[1, 4], [1, 2], [1, 4]]], [8, [7, 8, 9], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[7, 9, 2, 2, 6, 6, Yes], [8, 8, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[7, 9, 2, 2, 6, 6, Yes], [8, 8, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
71. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[8, [8, 11, 14], [[1, 4], [1, 2], [1, 4]]], [9, [8, 9, 10], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[8, 11, 3, 9/2, 27/2, 27/2, Yes], [9, 9, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[8, 11, 3, 9/2, 27/2, 27/2, Yes], [9, 9, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
72. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[9, [9, 10, 11], [[1, 4], [1, 2], [1, 4]]], [10, [9, 10, 11], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[10, 10, 0, 1/2, 1/2, 1/2, Yes], [9, 10, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[10, 10, 0, 1/2, 1/2, 1/2, Yes], [9, 10, 1, 1/2, 3/2, 3/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
73. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[10, [10, 12, 14], [[1, 4], [1, 2], [1, 4]]], [11, [10, 11, 12], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[10, 12, 2, 2, 6, 6, Yes], [11, 11, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[10, 12, 2, 2, 6, 6, Yes], [11, 11, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
74. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[11, [11, 14, 17], [[1, 4], [1, 2], [1, 4]]], [12, [11, 12, 13], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[11, 14, 3, 9/2, 27/2, 27/2, Yes], [12, 12, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[11, 14, 3, 9/2, 27/2, 27/2, Yes], [12, 12, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
75. Bias-Variance Decomposition Is Checked Pointwise under Each Theta Law
Use complete laws=[[12, [12, 13, 14], [[1, 4], [1, 2], [1, 4]]], [13, [12, 13, 14], [[1, 4], [1, 2], [1, 4]]]], each [theta,estimator values,aligned probabilities]. Every exact law is separately normalized and theta labels are unique. Give (theta_identity_rows, all_identities_hold, certificate).
View answer and explanation
([[12, 13, 1, 1/2, 3/2, 3/2, Yes], [13, 13, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim)
For each law separately retain [theta,mean,bias,variance,MSE,variance+bias^2,identity]. Preserve the source canonical law order rather than numeric theta order. Do not pool laws or infer asymptotic behavior from a finite table. The exact result, retaining the requested order, is ([[12, 13, 1, 1/2, 3/2, 3/2, Yes], [13, 13, 0, 1/2, 1/2, 1/2, Yes]], Yes, identity_is_pointwise_in_theta_not_an_asymptotic_claim).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-multi-theta-table · candidate
76. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-12, complete estimator values=[-13, -12, -11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-12, -13, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-12, -13, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
77. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-11, complete estimator values=[-11, -9, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-11. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-11, -11, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-11, -11, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
78. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-10, complete estimator values=[-13, -10, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-13. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-10, -13, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-10, -13, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
79. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-9, complete estimator values=[-9, -8, -7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-9. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-9, -9, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-9, -9, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
80. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-8, complete estimator values=[-10, -8, -6] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-10. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-8, -10, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-8, -10, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
81. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-7, complete estimator values=[-7, -4, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-7. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-7, -7, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-7, -7, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
82. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-6, complete estimator values=[-7, -6, -5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-7. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-6, -7, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-6, -7, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
83. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-5, complete estimator values=[-5, -3, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-5. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-5, -5, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-5, -5, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
84. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-4, complete estimator values=[-7, -4, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-7. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-4, -7, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-4, -7, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
85. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-3, complete estimator values=[-3, -2, -1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-3. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-3, -3, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-3, -3, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
86. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-2, complete estimator values=[-4, -2, 0] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-4. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-2, -4, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-2, -4, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
87. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=-1, complete estimator values=[-1, 2, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-1. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(-1, -1, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (-1, -1, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
88. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=0, complete estimator values=[-1, 0, 1] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-1. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(0, -1, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (0, -1, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
89. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=1, complete estimator values=[1, 3, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=1. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(1, 1, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (1, 1, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
90. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=2, complete estimator values=[-1, 2, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=-1. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(2, -1, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (2, -1, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
91. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=3, complete estimator values=[3, 4, 5] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=3. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(3, 3, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (3, 3, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
92. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=4, complete estimator values=[2, 4, 6] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=2. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(4, 2, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (4, 2, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
93. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=5, complete estimator values=[5, 8, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=5. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(5, 5, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (5, 5, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
94. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=6, complete estimator values=[5, 6, 7] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=5. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(6, 5, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (6, 5, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
95. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=7, complete estimator values=[7, 9, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=7. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(7, 7, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (7, 7, 0, 2, No, 2, 6, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
96. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=8, complete estimator values=[5, 8, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=5. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(8, 5, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (8, 5, 9, 0, Yes, 9/2, 9/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
97. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=9, complete estimator values=[9, 10, 11] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=9. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(9, 9, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (9, 9, 0, 1, No, 1/2, 3/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
98. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=10, complete estimator values=[8, 10, 12] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=8. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(10, 8, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (10, 8, 4, 0, Yes, 2, 2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
99. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=11, complete estimator values=[11, 14, 17] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=11. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(11, 11, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (11, 11, 0, 3, No, 9/2, 27/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
100. Unbiasedness Gives MSE Equal to Variance, Not Zero Error or Consistency
Use scalar theta=12, complete estimator values=[11, 12, 13] and aligned exact probabilities=[1/4, 1/2, 1/4]. Under this same P_theta the nonnegative probabilities sum to1. Retain zero-mass listed rows and sort values with their own probabilities. Given listed observed estimate=11. Give (theta, observed, observed_squared_error, bias, unbiased, variance, MSE, variance_comparison_holds, unbiased_and_zero_MSE, consistency_claim, boundary).
variance_comparison_holds checks MSE=variance when unbiased, otherwise MSE≥variance. One observed loss may coincide with MSE; the final No does not deny coincidence or establish consistency.
View answer and explanation
(12, 11, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE)
Compute the observed squared error separately from expected loss. If unbiased, verify MSE=variance; otherwise verify MSE≥variance. The last No withholds consistency, not possible coincidence between one observed loss and MSE. Unbiasedness may coexist with positive variance and MSE. The exact result, retaining the requested order, is (12, 11, 1, 0, Yes, 1/2, 1/2, Yes, No, No, unbiased_only_gives_MSE_equals_variance_not_zero_MSE_or_consistency_and_observed_error_is_not_MSE).
Reviewed step-by-step derivation
Bias-variance decomposition · qf-mod46-biasvar-biased-unbiased-boundary · candidate
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