# Parameter-Indexed Expectations, Bias and Full-Model Unbiasedness

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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.

batch-20261014-advance-20261106-r03-estimator-bias-foundations · 2026-10-14 · 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.

[Field-order and source-binding data](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 1. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 2, 3/2, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0001/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 2. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 1/2, 13/5, 21/10, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0002/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 3. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 11/3, 19/6, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0003/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 4. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 2/3, 71/17, 179/51, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0004/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 5. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 38/7, 69/14, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0005/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 6. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 3/4, 111/19, 387/76, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0006/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 7. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 29/4, 27/4, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0007/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 8. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 4/5, 53/7, 237/35, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0008/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 9. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 82/9, 155/18, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0009/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 10. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 5/6, 215/23, 1175/138, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0010/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 11. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 11, 21/2, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0011/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 12. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 6/7, 279/25, 1803/175, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0012/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 13. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 142/11, 273/22, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0013/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 14. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 7/8, 13, 97/8, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0014/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 15. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 89/6, 43/3, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0015/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 16. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 8/9, 431/29, 3647/261, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0016/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 17. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 218/13, 423/26, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0017/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 18. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 9/10, 519/31, 4911/310, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0018/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 19. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 131/7, 255/14, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0019/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 20. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 10/11, 205/11, 195/11, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0020/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 21. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 62/3, 121/6, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0021/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 22. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 11/12, 719/35, 8243/420, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0022/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 23. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 181/8, 177/8, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0023/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 24. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (1, 12/13, 831/37, 10359/481, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0024/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 25. Subtract the Target from the Expectation under That Same Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, theta_value, expectation, bias, certificate).

View answer and explanation(0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution)

Compute the full weighted expectation under that row, then subtract its aligned target theta. An observed estimate or another parameter law is not the bias at this parameter. The exact result, retaining the requested order, is (0, 1/2, 418/17, 819/34, expectation_from_same_theta_sampling_distribution).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0025/)

Estimator bias · qf-mod37-bias-at-parameter · candidate





## 26. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 2, 3/2], [1, 1/3, 13/7, 32/21]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0026/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 27. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 31/11, 51/22], [1, 1/2, 13/5, 21/10]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0027/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 28. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 11/3, 19/6], [1, 3/5, 27/8, 111/40]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0028/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 29. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 59/13, 105/26], [1, 2/3, 71/17, 179/51]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0029/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 30. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 38/7, 69/14], [1, 5/7, 5, 30/7]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0030/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 31. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 19/3, 35/6], [1, 3/4, 111/19, 387/76]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0031/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 32. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 29/4, 27/4], [1, 7/9, 67/10, 533/90]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0032/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 33. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 139/17, 261/34], [1, 4/5, 53/7, 237/35]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0033/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 34. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 82/9, 155/18], [1, 9/11, 93/11, 84/11]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0034/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 35. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 191/19, 363/38], [1, 5/6, 215/23, 1175/138]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0035/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 36. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 11, 21/2], [1, 11/13, 41/4, 489/52]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0036/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 37. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 251/21, 481/42], [1, 6/7, 279/25, 1803/175]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0037/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 38. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 142/11, 273/22], [1, 13/15, 157/13, 2186/195]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0038/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 39. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 319/23, 615/46], [1, 7/8, 13, 97/8]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0039/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 40. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 89/6, 43/3], [1, 15/17, 195/14, 3105/238]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0040/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 41. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 79/5, 153/10], [1, 8/9, 431/29, 3647/261]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0041/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 42. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 218/13, 423/26], [1, 17/19, 79/5, 1416/95]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0042/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 43. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 479/27, 931/54], [1, 9/10, 519/31, 4911/310]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0043/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 44. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 131/7, 255/14], [1, 19/21, 283/16, 5639/336]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0044/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 45. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 571/29, 1113/58], [1, 10/11, 205/11, 195/11]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0045/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 46. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 62/3, 121/6], [1, 21/23, 333/17, 7302/391]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0046/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 47. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 671/31, 1311/62], [1, 11/12, 719/35, 8243/420]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0047/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 48. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 181/8, 177/8], [1, 23/25, 43/2, 1029/50]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0048/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 49. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 779/33, 1525/66], [1, 12/13, 831/37, 10359/481]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0049/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 50. Bias Is an Indexed Family, Not One Observed Error

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_bias_rows, boundary).

View answer and explanation([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family)

For every parameter index increasingly, report the aligned target, exact expectation and expectation-minus-target. Do not average biases over parameters or introduce a prior. The exact result, retaining the requested order, is ([[0, 1/2, 418/17, 819/34], [1, 25/27, 445/19, 11540/513]], bias_is_parameter_indexed_family).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0050/)

Estimator bias · qf-mod37-bias-family-readback · candidate





## 51. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 3/2], [1, 32/21]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 3/2], [1, 32/21]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0051/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 52. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 51/22], [1, 21/10]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 51/22], [1, 21/10]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0052/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 53. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 19/6], [1, 111/40]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0053/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 54. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 105/26], [1, 179/51]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 105/26], [1, 179/51]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0054/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 55. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 69/14], [1, 30/7]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 69/14], [1, 30/7]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0055/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 56. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 35/6], [1, 387/76]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 35/6], [1, 387/76]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0056/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 57. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 27/4], [1, 533/90]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 27/4], [1, 533/90]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0057/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 58. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 261/34], [1, 237/35]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 261/34], [1, 237/35]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0058/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 59. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 155/18], [1, 84/11]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 155/18], [1, 84/11]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0059/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 60. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 363/38], [1, 1175/138]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 363/38], [1, 1175/138]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0060/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 61. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 21/2], [1, 489/52]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 21/2], [1, 489/52]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0061/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 62. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 481/42], [1, 1803/175]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 481/42], [1, 1803/175]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0062/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 63. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 273/22], [1, 2186/195]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 273/22], [1, 2186/195]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0063/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 64. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 615/46], [1, 97/8]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 615/46], [1, 97/8]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0064/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 65. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 43/3], [1, 3105/238]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 43/3], [1, 3105/238]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0065/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 66. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 153/10], [1, 3647/261]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 153/10], [1, 3647/261]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0066/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 67. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 423/26], [1, 1416/95]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 423/26], [1, 1416/95]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0067/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 68. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 931/54], [1, 4911/310]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 931/54], [1, 4911/310]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0068/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 69. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 255/14], [1, 5639/336]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 255/14], [1, 5639/336]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0069/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 70. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 1113/58], [1, 195/11]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1113/58], [1, 195/11]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0070/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 71. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 121/6], [1, 7302/391]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 121/6], [1, 7302/391]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0071/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 72. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 1311/62], [1, 8243/420]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1311/62], [1, 8243/420]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0072/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 73. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 177/8], [1, 1029/50]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 177/8], [1, 1029/50]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0073/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 74. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 1525/66], [1, 10359/481]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 1525/66], [1, 10359/481]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0074/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 75. Unbiasedness Requires Zero Bias at Every Declared Parameter

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_biases, unbiased_all_parameters, boundary).

View answer and explanation([[0, 819/34], [1, 11540/513]], No, unbiased_requires_every_theta)

Compute every exact bias and require all of them to vanish. Zero bias at one parameter, or one estimate matching a target, does not establish unbiasedness over this complete finite model or any unprovided parameter space. The exact result, retaining the requested order, is ([[0, 819/34], [1, 11540/513]], No, unbiased_requires_every_theta).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0075/)

Estimator bias · qf-mod37-bias-unbiased-audit · candidate





## 76. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 1/3], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 3/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 3/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0076/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 77. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 2/4], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 1, 1, 21/10, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 21/10, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0077/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 78. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 3/5], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 2, 4, 19/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 4, 19/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0078/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 79. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 4/6], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 3, 6, 179/51, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 6, 179/51, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0079/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 80. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 5/7], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 69/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 69/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0080/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 81. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 6/8], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 1, 1, 387/76, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 387/76, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0081/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 82. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 7/9], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 2, 8, 27/4, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 8, 27/4, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0082/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 83. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 8/10], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 3, 10, 237/35, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 10, 237/35, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0083/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 84. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 9/11], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 155/18, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 155/18, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0084/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 85. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 10/12], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 1, 1, 1175/138, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 1175/138, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0085/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 86. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 11/13], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 2, 12, 21/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 12, 21/2, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0086/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 87. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 12/14], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 3, 14, 1803/175, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 14, 1803/175, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0087/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 88. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 13/15], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 273/22, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0088/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 89. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 14/16], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 1, 1, 97/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 97/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0089/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 90. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 15/17], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 2, 16, 43/3, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 16, 43/3, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0090/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 91. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 16/18], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 3, 18, 3647/261, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 18, 3647/261, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0091/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 92. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 17/19], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 423/26, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 423/26, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0092/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 93. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 18/20], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 1, 1, 4911/310, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 4911/310, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0093/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 94. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 19/21], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 2, 20, 255/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 20, 255/14, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0094/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 95. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 20/22], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 3, 22, 195/11, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 22, 195/11, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0095/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 96. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 21/23], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 121/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 121/6, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0096/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 97. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 22/24], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=1. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 1, 1, 8243/420, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 1, 1, 8243/420, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0097/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 98. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 23/25], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=2. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 2, 24, 177/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 2, 24, 177/8, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0098/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 99. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 24/26], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1, supplied outcome=3. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(1, 3, 26, 10359/481, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (1, 3, 26, 10359/481, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0099/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate





## 100. One Realized Estimate Does Not Establish Bias or Other Properties

Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. The aligned parameter values are theta_values=[1/2, 25/27], and the target is g(theta)=theta. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0, supplied outcome=0. Give (parameter_index, outcome, estimate, bias, observed_value_suffices_for_property_claims, boundary).

No withholds statistical-property claims from one observation; it is not a numerical test of whether this estimate equals the bias.

View answer and explanation(0, 0, 0, 819/34, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE)

Read T at that outcome and retain the separate exact bias under the selected law. No withholds other statistical-property claims from one observation; it does not assert that the estimate and bias have unequal numerical values. Unbiasedness alone does not supply zero variance, consistency, sufficiency or minimum MSE. The exact result, retaining the requested order, is (0, 0, 0, 819/34, No, observed_estimate_not_bias_and_unbiased_not_zero_variance_consistency_sufficiency_or_minimum_MSE).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261106-r03-estimator-bias-0100/)

Estimator bias · qf-mod37-bias-observed-boundary · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r03-estimator-bias-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261106-r03-estimator-bias-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261106-r03-estimator-bias-foundations/validation-report.json)