Same-Law Central Moments, Second-Moment Identities and Nonclaims
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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-r04-estimator-variance-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
1. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 2, 1, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 2, 1, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
2. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 13/5, 56/25, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 13/5, 56/25, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
3. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 11/3, 29/9, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 11/3, 29/9, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
4. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 71/17, 1606/289, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 71/17, 1606/289, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
5. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 38/7, 313/49, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 38/7, 313/49, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
6. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 111/19, 3620/361, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 111/19, 3620/361, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
7. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 29/4, 163/16, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 29/4, 163/16, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
8. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 53/7, 754/49, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 53/7, 754/49, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
9. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 82/9, 1169/81, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 82/9, 1169/81, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
10. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 215/23, 11344/529, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 215/23, 11344/529, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
11. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 11, 19, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 11, 19, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
12. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 279/25, 17534/625, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 279/25, 17534/625, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
13. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 142/11, 2881/121, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 142/11, 2881/121, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
14. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 13, 316/9, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 13, 316/9, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
15. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 89/6, 1037/36, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 89/6, 1037/36, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
16. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 431/29, 35770/841, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 431/29, 35770/841, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
17. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 218/13, 5737/169, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 218/13, 5737/169, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
18. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 519/31, 48296/961, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 519/31, 48296/961, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
19. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 131/7, 1921/49, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 131/7, 1921/49, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
20. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 205/11, 7046/121, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 205/11, 7046/121, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
21. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 62/3, 401/9, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 62/3, 401/9, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
22. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 719/35, 81364/1225, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 719/35, 81364/1225, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
23. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 181/8, 3199/64, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 181/8, 3199/64, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
24. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(1, 831/37, 102386/1369, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (1, 831/37, 102386/1369, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
25. Use One Parameter Law and Its Own Mean for Variance
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, expectation, variance, variance_certificate, certificate).
View answer and explanation
(0, 418/17, 16033/289, Yes, same_theta_sampling_law_support_and_mean)
Compute E_theta[T] and E_theta[(T−E_theta[T])^2] exactly under that same law. A zero variance implies constancy almost surely under that law, not equality with a target or constancy on zero-mass table entries. The exact result, retaining the requested order, is (0, 418/17, 16033/289, Yes, same_theta_sampling_law_support_and_mean).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-at-parameter · candidate
26. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 2, 1], [1, 13/7, 55/49]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 2, 1], [1, 13/7, 55/49]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
27. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 31/11, 238/121], [1, 13/5, 56/25]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
28. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 11/3, 29/9], [1, 27/8, 239/64]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 11/3, 29/9], [1, 27/8, 239/64]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
29. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 59/13, 796/169], [1, 71/17, 1606/289]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 59/13, 796/169], [1, 71/17, 1606/289]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
30. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 38/7, 313/49], [1, 5, 23/3]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
31. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 19/3, 74/9], [1, 111/19, 3620/361]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 19/3, 74/9], [1, 111/19, 3620/361]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
32. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 29/4, 163/16], [1, 67/10, 1261/100]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 29/4, 163/16], [1, 67/10, 1261/100]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
33. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 139/17, 3544/289], [1, 53/7, 754/49]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 139/17, 3544/289], [1, 53/7, 754/49]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
34. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 82/9, 1169/81], [1, 93/11, 2219/121]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 82/9, 1169/81], [1, 93/11, 2219/121]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
35. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 191/19, 6022/361], [1, 215/23, 11344/529]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 191/19, 6022/361], [1, 215/23, 11344/529]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
36. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 11, 19], [1, 41/4, 395/16]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 11, 19], [1, 41/4, 395/16]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
37. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 251/21, 9428/441], [1, 279/25, 17534/625]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 251/21, 9428/441], [1, 279/25, 17534/625]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
38. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 142/11, 2881/121], [1, 157/13, 5329/169]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 142/11, 2881/121], [1, 157/13, 5329/169]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
39. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 319/23, 13906/529], [1, 13, 316/9]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
40. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 89/6, 1037/36], [1, 195/14, 7601/196]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 89/6, 1037/36], [1, 195/14, 7601/196]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
41. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 79/5, 784/25], [1, 431/29, 35770/841]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 79/5, 784/25], [1, 431/29, 35770/841]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
42. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 218/13, 5737/169], [1, 79/5, 1159/25]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 218/13, 5737/169], [1, 79/5, 1159/25]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
43. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 479/27, 26654/729], [1, 519/31, 48296/961]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 479/27, 26654/729], [1, 519/31, 48296/961]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
44. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 131/7, 1921/49], [1, 283/16, 13879/256]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 131/7, 1921/49], [1, 283/16, 13879/256]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
45. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 571/29, 35212/841], [1, 205/11, 7046/121]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 571/29, 35212/841], [1, 205/11, 7046/121]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
46. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 62/3, 401/9], [1, 333/17, 18005/289]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 62/3, 401/9], [1, 333/17, 18005/289]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
47. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 671/31, 45418/961], [1, 719/35, 81364/1225]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 671/31, 45418/961], [1, 719/35, 81364/1225]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
48. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 181/8, 3199/64], [1, 43/2, 847/12]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 181/8, 3199/64], [1, 43/2, 847/12]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
49. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 779/33, 57416/1089], [1, 831/37, 102386/1369]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
50. Estimator Variance Remains Parameter-Indexed
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variance_rows, certificate).
View answer and explanation
([[0, 418/17, 16033/289], [1, 445/19, 28531/361]], variance_is_parameter_indexed_family)
For each parameter index increasingly, give its expectation and exact variance. Keep the law, positive-mass support and mean bound to that same index; no bias or prior-weighted averaging is inferred. The exact result, retaining the requested order, is ([[0, 418/17, 16033/289], [1, 445/19, 28531/361]], variance_is_parameter_indexed_family).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-family-readback · candidate
51. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 5, 2, 1, 1, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 5, 2, 1, 1, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
52. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 9, 13/5, 56/25, 56/25, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 9, 13/5, 56/25, 56/25, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
53. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 50/3, 11/3, 29/9, 29/9, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 50/3, 11/3, 29/9, 29/9, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
54. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 23, 71/17, 1606/289, 1606/289, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 23, 71/17, 1606/289, 1606/289, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
55. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 251/7, 38/7, 313/49, 313/49, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 251/7, 38/7, 313/49, 313/49, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
56. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 839/19, 111/19, 3620/361, 3620/361, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 839/19, 111/19, 3620/361, 3620/361, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
57. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 251/4, 29/4, 163/16, 163/16, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 251/4, 29/4, 163/16, 163/16, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
58. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 509/7, 53/7, 754/49, 754/49, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 509/7, 53/7, 754/49, 754/49, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
59. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 877/9, 82/9, 1169/81, 1169/81, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 877/9, 82/9, 1169/81, 1169/81, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
60. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 2503/23, 215/23, 11344/529, 11344/529, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 2503/23, 215/23, 11344/529, 11344/529, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
61. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 140, 11, 19, 19, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 140, 11, 19, 19, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
62. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 763/5, 279/25, 17534/625, 17534/625, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 763/5, 279/25, 17534/625, 17534/625, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
63. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 2095/11, 142/11, 2881/121, 2881/121, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 2095/11, 142/11, 2881/121, 2881/121, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
64. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 1837/9, 13, 316/9, 316/9, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 1837/9, 13, 316/9, 316/9, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
65. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 1493/6, 89/6, 1037/36, 1037/36, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 1493/6, 89/6, 1037/36, 1037/36, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
66. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 7639/29, 431/29, 35770/841, 35770/841, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 7639/29, 431/29, 35770/841, 35770/841, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
67. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 4097/13, 218/13, 5737/169, 5737/169, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 4097/13, 218/13, 5737/169, 5737/169, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
68. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 10247/31, 519/31, 48296/961, 48296/961, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 10247/31, 519/31, 48296/961, 48296/961, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
69. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 2726/7, 131/7, 1921/49, 1921/49, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 2726/7, 131/7, 1921/49, 1921/49, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
70. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 4461/11, 205/11, 7046/121, 7046/121, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 4461/11, 205/11, 7046/121, 7046/121, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
71. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 1415/3, 62/3, 401/9, 401/9, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 1415/3, 62/3, 401/9, 401/9, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
72. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 3419/7, 719/35, 81364/1225, 81364/1225, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 3419/7, 719/35, 81364/1225, 81364/1225, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
73. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 4495/8, 181/8, 3199/64, 3199/64, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 4495/8, 181/8, 3199/64, 3199/64, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
74. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=1. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(1, 21431/37, 831/37, 102386/1369, 102386/1369, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (1, 21431/37, 831/37, 102386/1369, 102386/1369, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
75. The Second Moment and Squared Mean Are Different Quantities
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Selected parameter index=0. Give (parameter_index, second_moment, expectation, second_minus_squared_mean, variance, equal).
View answer and explanation
(0, 11221/17, 418/17, 16033/289, 16033/289, Yes)
Compute E[T^2] and E[T] separately, then subtract the square of the mean from the second moment. Retain both equal exact variance fields and the identity flag; do not replace E[T^2] by (E[T])^2. The exact result, retaining the requested order, is (0, 11221/17, 418/17, 16033/289, 16033/289, Yes).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-second-moment-identity · candidate
76. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/10, 2/10, 3/10, 4/10], [2/14, 3/14, 4/14, 5/14]], and one parameter-independent total statistic T=[0, 1, 2, 3] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 1], [1, 55/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1], [1, 55/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
77. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/11, 2/11, 3/11, 5/11], [2/15, 3/15, 4/15, 6/15]], and one parameter-independent total statistic T=[0, 1, 3, 4] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 238/121], [1, 56/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 238/121], [1, 56/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
78. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/12, 2/12, 3/12, 6/12], [2/16, 3/16, 4/16, 7/16]], and one parameter-independent total statistic T=[0, 1, 4, 5] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 29/9], [1, 239/64]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 29/9], [1, 239/64]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
79. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/13, 2/13, 3/13, 7/13], [2/17, 3/17, 4/17, 8/17]], and one parameter-independent total statistic T=[0, 1, 5, 6] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 796/169], [1, 1606/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 796/169], [1, 1606/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
80. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/14, 2/14, 3/14, 8/14], [2/18, 3/18, 4/18, 9/18]], and one parameter-independent total statistic T=[0, 1, 6, 7] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 313/49], [1, 23/3]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 313/49], [1, 23/3]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
81. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/15, 2/15, 3/15, 9/15], [2/19, 3/19, 4/19, 10/19]], and one parameter-independent total statistic T=[0, 1, 7, 8] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 74/9], [1, 3620/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 74/9], [1, 3620/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
82. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/16, 2/16, 3/16, 10/16], [2/20, 3/20, 4/20, 11/20]], and one parameter-independent total statistic T=[0, 1, 8, 9] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 163/16], [1, 1261/100]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 163/16], [1, 1261/100]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
83. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/17, 2/17, 3/17, 11/17], [2/21, 3/21, 4/21, 12/21]], and one parameter-independent total statistic T=[0, 1, 9, 10] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 3544/289], [1, 754/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 3544/289], [1, 754/49]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
84. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/18, 2/18, 3/18, 12/18], [2/22, 3/22, 4/22, 13/22]], and one parameter-independent total statistic T=[0, 1, 10, 11] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 1169/81], [1, 2219/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1169/81], [1, 2219/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
85. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/19, 2/19, 3/19, 13/19], [2/23, 3/23, 4/23, 14/23]], and one parameter-independent total statistic T=[0, 1, 11, 12] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 6022/361], [1, 11344/529]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 6022/361], [1, 11344/529]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
86. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/20, 2/20, 3/20, 14/20], [2/24, 3/24, 4/24, 15/24]], and one parameter-independent total statistic T=[0, 1, 12, 13] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 19], [1, 395/16]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 19], [1, 395/16]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
87. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/21, 2/21, 3/21, 15/21], [2/25, 3/25, 4/25, 16/25]], and one parameter-independent total statistic T=[0, 1, 13, 14] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 9428/441], [1, 17534/625]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 9428/441], [1, 17534/625]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
88. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/22, 2/22, 3/22, 16/22], [2/26, 3/26, 4/26, 17/26]], and one parameter-independent total statistic T=[0, 1, 14, 15] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 2881/121], [1, 5329/169]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 2881/121], [1, 5329/169]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
89. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/23, 2/23, 3/23, 17/23], [2/27, 3/27, 4/27, 18/27]], and one parameter-independent total statistic T=[0, 1, 15, 16] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 13906/529], [1, 316/9]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 13906/529], [1, 316/9]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
90. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/24, 2/24, 3/24, 18/24], [2/28, 3/28, 4/28, 19/28]], and one parameter-independent total statistic T=[0, 1, 16, 17] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 1037/36], [1, 7601/196]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1037/36], [1, 7601/196]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
91. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/25, 2/25, 3/25, 19/25], [2/29, 3/29, 4/29, 20/29]], and one parameter-independent total statistic T=[0, 1, 17, 18] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 784/25], [1, 35770/841]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 784/25], [1, 35770/841]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
92. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/26, 2/26, 3/26, 20/26], [2/30, 3/30, 4/30, 21/30]], and one parameter-independent total statistic T=[0, 1, 18, 19] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 5737/169], [1, 1159/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 5737/169], [1, 1159/25]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
93. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/27, 2/27, 3/27, 21/27], [2/31, 3/31, 4/31, 22/31]], and one parameter-independent total statistic T=[0, 1, 19, 20] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 26654/729], [1, 48296/961]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 26654/729], [1, 48296/961]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
94. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/28, 2/28, 3/28, 22/28], [2/32, 3/32, 4/32, 23/32]], and one parameter-independent total statistic T=[0, 1, 20, 21] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 1921/49], [1, 13879/256]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 1921/49], [1, 13879/256]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
95. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/29, 2/29, 3/29, 23/29], [2/33, 3/33, 4/33, 24/33]], and one parameter-independent total statistic T=[0, 1, 21, 22] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 35212/841], [1, 7046/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 35212/841], [1, 7046/121]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
96. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/30, 2/30, 3/30, 24/30], [2/34, 3/34, 4/34, 25/34]], and one parameter-independent total statistic T=[0, 1, 22, 23] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 401/9], [1, 18005/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 401/9], [1, 18005/289]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
97. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/31, 2/31, 3/31, 25/31], [2/35, 3/35, 4/35, 26/35]], and one parameter-independent total statistic T=[0, 1, 23, 24] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 45418/961], [1, 81364/1225]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 45418/961], [1, 81364/1225]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
98. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/32, 2/32, 3/32, 26/32], [2/36, 3/36, 4/36, 27/36]], and one parameter-independent total statistic T=[0, 1, 24, 25] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 3199/64], [1, 847/12]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 3199/64], [1, 847/12]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
99. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/33, 2/33, 3/33, 27/33], [2/37, 3/37, 4/37, 28/37]], and one parameter-independent total statistic T=[0, 1, 25, 26] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 57416/1089], [1, 102386/1369]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 57416/1089], [1, 102386/1369]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
100. Variance Alone Withholds Unbiasedness, Consistency and MSE Claims
Use indexed parameter slots Fin(2)={0,…,1}, complete exact rational law rows=[[1/34, 2/34, 3/34, 28/34], [2/38, 3/38, 4/38, 29/38]], and one parameter-independent total statistic T=[0, 1, 26, 27] in common outcome order. Every row is nonnegative and sums to1. Keep all expectations and moments under the same selected law; zero-mass outcomes remain in the table but do not enlarge its positive-mass support. Give (parameter_variances, claim_marker_1, claim_marker_2, claim_marker_3, boundary).
The three No values withhold unbiasedness, consistency or MSE claims; they do not prove the corresponding properties false. Zero variance means almost-sure constancy only under the same parameter law.
View answer and explanation
([[0, 16033/289], [1, 28531/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE)
List all parameter variances, then the three literal No nonclaim markers. They do not prove corresponding statistical properties false. Consistency needs a sequence of models/statistics, while MSE also needs the target/bias; zero variance gives only same-law almost-sure constancy. The exact result, retaining the requested order, is ([[0, 16033/289], [1, 28531/361]], No, No, No, zero_variance_only_same_theta_almost_sure_constant_not_unbiased_consistent_or_MSE).
Reviewed step-by-step derivation
Estimator variance · qf-mod37-estvar-property-boundary · candidate
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