Declared IID Premises, Chebyshev Upper Bounds and Strict All-n Tails
This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.
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-20261105-r07-weak-law-large-numbers-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. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=5. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(0, 1/2, 5, 1/10, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (0, 1/2, 5, 1/10, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
2. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=6. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(1/2, 2/3, 6, 1/9, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (1/2, 2/3, 6, 1/9, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
3. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=7. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(2/3, 3/4, 7, 3/28, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (2/3, 3/4, 7, 3/28, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
4. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=8. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(3/4, 4/5, 8, 1/10, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (3/4, 4/5, 8, 1/10, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
5. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=9. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(4/5, 5/6, 9, 5/54, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (4/5, 5/6, 9, 5/54, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
6. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=10. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(5/6, 6/7, 10, 3/35, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (5/6, 6/7, 10, 3/35, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
7. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=11. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(6/7, 7/8, 11, 7/88, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (6/7, 7/8, 11, 7/88, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
8. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=12. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(7/8, 8/9, 12, 2/27, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (7/8, 8/9, 12, 2/27, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
9. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=13. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(8/9, 9/10, 13, 9/130, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (8/9, 9/10, 13, 9/130, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
10. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=14. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(9/10, 10/11, 14, 5/77, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (9/10, 10/11, 14, 5/77, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
11. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=15. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(10/11, 11/12, 15, 11/180, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (10/11, 11/12, 15, 11/180, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
12. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=16. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(11/12, 12/13, 16, 3/52, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (11/12, 12/13, 16, 3/52, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
13. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=17. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (12/13, 13/14, 17, 13/238, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
14. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=18. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(13/14, 14/15, 18, 7/135, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (13/14, 14/15, 18, 7/135, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
15. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=19. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(14/15, 15/16, 19, 15/304, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (14/15, 15/16, 19, 15/304, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
16. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=20. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(15/16, 16/17, 20, 4/85, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (15/16, 16/17, 20, 4/85, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
17. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=21. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(16/17, 17/18, 21, 17/378, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (16/17, 17/18, 21, 17/378, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
18. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=22. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(17/18, 18/19, 22, 9/209, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (17/18, 18/19, 22, 9/209, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
19. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=23. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(18/19, 19/20, 23, 19/460, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (18/19, 19/20, 23, 19/460, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
20. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=24. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(19/20, 20/21, 24, 5/126, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (19/20, 20/21, 24, 5/126, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
21. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=25. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(20/21, 21/22, 25, 21/550, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (20/21, 21/22, 25, 21/550, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
22. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=26. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(21/22, 22/23, 26, 11/299, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (21/22, 22/23, 26, 11/299, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
23. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=27. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(22/23, 23/24, 27, 23/648, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (22/23, 23/24, 27, 23/648, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
24. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=28. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(23/24, 24/25, 28, 6/175, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (23/24, 24/25, 28, 6/175, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
25. Independence Gives Variance of the Average, Not a Finite-Sample Limit
Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive sample size n=29. Give (mean, variance, n, average_variance, certificate).
View answer and explanation
(24/25, 25/26, 29, 25/754, independence_gives_variance_sigma_squared_over_n)
Use the stated IID premise to divide the common variance by n. This is an exact conditional variance identity, not evidence that a particular observed sample mean equals its expectation. The exact result, retaining the requested order, is (24/25, 25/26, 29, 25/754, independence_gives_variance_sigma_squared_over_n).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-variance-average · candidate
26. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/2 and sample size n=20. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(0, 1/2, 20, 1/10, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (0, 1/2, 20, 1/10, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
27. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/3 and sample size n=21. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(1/2, 1/3, 21, 2/7, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (1/2, 1/3, 21, 2/7, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
28. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/4 and sample size n=22. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(2/3, 1/4, 22, 6/11, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (2/3, 1/4, 22, 6/11, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
29. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/5 and sample size n=23. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(3/4, 1/5, 23, 20/23, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (3/4, 1/5, 23, 20/23, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
30. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/6 and sample size n=24. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(4/5, 1/6, 24, 5/4, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (4/5, 1/6, 24, 5/4, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
31. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/7 and sample size n=25. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(5/6, 1/7, 25, 42/25, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (5/6, 1/7, 25, 42/25, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
32. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/8 and sample size n=26. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(6/7, 1/8, 26, 28/13, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (6/7, 1/8, 26, 28/13, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
33. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/9 and sample size n=27. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(7/8, 1/9, 27, 8/3, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (7/8, 1/9, 27, 8/3, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
34. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/10 and sample size n=28. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(8/9, 1/10, 28, 45/14, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (8/9, 1/10, 28, 45/14, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
35. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/11 and sample size n=29. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(9/10, 1/11, 29, 110/29, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (9/10, 1/11, 29, 110/29, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
36. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/12 and sample size n=30. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(10/11, 1/12, 30, 22/5, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (10/11, 1/12, 30, 22/5, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
37. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/13 and sample size n=31. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(11/12, 1/13, 31, 156/31, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (11/12, 1/13, 31, 156/31, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
38. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/14 and sample size n=32. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(12/13, 1/14, 32, 91/16, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (12/13, 1/14, 32, 91/16, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
39. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/15 and sample size n=33. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(13/14, 1/15, 33, 70/11, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (13/14, 1/15, 33, 70/11, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
40. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16 and sample size n=34. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(14/15, 1/16, 34, 120/17, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (14/15, 1/16, 34, 120/17, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
41. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/17 and sample size n=35. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(15/16, 1/17, 35, 272/35, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (15/16, 1/17, 35, 272/35, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
42. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/18 and sample size n=36. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(16/17, 1/18, 36, 17/2, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (16/17, 1/18, 36, 17/2, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
43. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/19 and sample size n=37. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(17/18, 1/19, 37, 342/37, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (17/18, 1/19, 37, 342/37, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
44. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/20 and sample size n=38. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(18/19, 1/20, 38, 10, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (18/19, 1/20, 38, 10, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
45. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/21 and sample size n=39. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(19/20, 1/21, 39, 140/13, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (19/20, 1/21, 39, 140/13, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
46. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/22 and sample size n=40. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(20/21, 1/22, 40, 231/20, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (20/21, 1/22, 40, 231/20, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
47. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/23 and sample size n=41. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(21/22, 1/23, 41, 506/41, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (21/22, 1/23, 41, 506/41, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
48. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/24 and sample size n=42. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(22/23, 1/24, 42, 92/7, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (22/23, 1/24, 42, 92/7, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
49. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/25 and sample size n=43. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(23/24, 1/25, 43, 600/43, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (23/24, 1/25, 43, 600/43, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
50. A Chebyshev Upper Bound May Exceed One without Being a Probability
Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26 and sample size n=44. Report the raw upper bound without clipping it to1. Give (mean, epsilon, n, probability_upper_bound, conditional_upper_bound_valid, certificate).
This is a Chebyshev upper bound conditional on the declared IID premise and may exceed1; it is not the event probability or a new Lean proof.
View answer and explanation
(24/25, 1/26, 44, 325/22, Yes, probability_upper_bound_not_finite_n_equality)
Compute variance/(n*epsilon^2). It is an upper bound, not the event probability or a finite-n equality claim; a bound above1 is loose but not an invalid probability value. The exact result, retaining the requested order, is (24/25, 1/26, 44, 325/22, Yes, probability_upper_bound_not_finite_n_equality).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-chebyshev-bound · candidate
51. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/2, delta=1/10. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/2, 1/10, 21, 2/21, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/2, 1/10, 21, 2/21, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
52. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/3, delta=1/11. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/3, 1/11, 67, 6/67, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/3, 1/11, 67, 6/67, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
53. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/4, delta=1/12. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/4, 1/12, 145, 12/145, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/4, 1/12, 145, 12/145, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
54. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/5, delta=1/13. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/5, 1/13, 261, 20/261, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/5, 1/13, 261, 20/261, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
55. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/6, delta=1/14. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/6, 1/14, 421, 30/421, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/6, 1/14, 421, 30/421, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
56. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/7, delta=1/15. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/7, 1/15, 631, 42/631, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/7, 1/15, 631, 42/631, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
57. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/8, delta=1/16. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/8, 1/16, 897, 56/897, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/8, 1/16, 897, 56/897, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
58. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/9, delta=1/17. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/9, 1/17, 1225, 72/1225, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/9, 1/17, 1225, 72/1225, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
59. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/10, delta=1/18. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/10, 1/18, 1621, 90/1621, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/10, 1/18, 1621, 90/1621, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
60. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/11, delta=1/19. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/11, 1/19, 2091, 110/2091, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/11, 1/19, 2091, 110/2091, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
61. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/12, delta=1/20. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/12, 1/20, 2641, 132/2641, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/12, 1/20, 2641, 132/2641, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
62. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/13, delta=1/21. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/13, 1/21, 3277, 156/3277, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/13, 1/21, 3277, 156/3277, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
63. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/14, delta=1/22. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/14, 1/22, 4005, 182/4005, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/14, 1/22, 4005, 182/4005, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
64. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/15, delta=1/23. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/15, 1/23, 4831, 210/4831, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/15, 1/23, 4831, 210/4831, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
65. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16, delta=1/24. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/16, 1/24, 5761, 240/5761, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/16, 1/24, 5761, 240/5761, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
66. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/17, delta=1/25. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/17, 1/25, 6801, 272/6801, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/17, 1/25, 6801, 272/6801, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
67. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/18, delta=1/26. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/18, 1/26, 7957, 306/7957, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/18, 1/26, 7957, 306/7957, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
68. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/19, delta=1/27. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/19, 1/27, 9235, 342/9235, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/19, 1/27, 9235, 342/9235, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
69. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/20, delta=1/28. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/20, 1/28, 10641, 380/10641, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/20, 1/28, 10641, 380/10641, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
70. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/21, delta=1/29. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/21, 1/29, 12181, 420/12181, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/21, 1/29, 12181, 420/12181, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
71. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/22, delta=1/30. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/22, 1/30, 13861, 462/13861, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/22, 1/30, 13861, 462/13861, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
72. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/23, delta=1/31. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/23, 1/31, 15687, 506/15687, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/23, 1/31, 15687, 506/15687, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
73. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/24, delta=1/32. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/24, 1/32, 17665, 552/17665, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/24, 1/32, 17665, 552/17665, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
74. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/25, delta=1/33. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/25, 1/33, 19801, 600/19801, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/25, 1/33, 19801, 600/19801, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
75. Choose a Strict All-n Chebyshev Tail Witness
Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=symbolic_all_finite_subfamilies. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26, delta=1/34. Give (epsilon, delta, N, chebyshev_bound_at_N, strict_delta_tail, certificate).
The IID marker declares a whole-sequence premise, not one proved by finite simulation or field labels; the strict delta bound must hold for every n≥N.
View answer and explanation
(1/26, 1/34, 22101, 650/22101, Yes, forall_n_ge_N_monotone_Chebyshev_tail)
Choose N=floor(variance/(delta*epsilon^2))+1 and compute variance/(N*epsilon^2). The strict bound is below delta, including at integral thresholds; monotonicity extends it to every n≥N under the same declared IID premise. The exact result, retaining the requested order, is (1/26, 1/34, 22101, 650/22101, Yes, forall_n_ge_N_monotone_Chebyshev_tail).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-epsilon-delta-witness · candidate
76. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=0/1, positive finite variance=1/2, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/2, finite prefix of sample means=[1/2, 2/3, 3/4]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(0, 1/2, [1/2, 2/3, 3/4], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (0, 1/2, [1/2, 2/3, 3/4], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
77. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=1/2, positive finite variance=2/3, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/3, finite prefix of sample means=[2/3, 3/4, 4/5]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(1/2, 1/3, [2/3, 3/4, 4/5], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (1/2, 1/3, [2/3, 3/4, 4/5], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
78. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=2/3, positive finite variance=3/4, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/4, finite prefix of sample means=[3/4, 4/5, 5/6]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(2/3, 1/4, [3/4, 4/5, 5/6], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (2/3, 1/4, [3/4, 4/5, 5/6], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
79. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=3/4, positive finite variance=4/5, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/5, finite prefix of sample means=[4/5, 5/6, 6/7]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(3/4, 1/5, [4/5, 5/6, 6/7], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (3/4, 1/5, [4/5, 5/6, 6/7], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
80. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=4/5, positive finite variance=5/6, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/6, finite prefix of sample means=[5/6, 6/7, 7/8]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(4/5, 1/6, [5/6, 6/7, 7/8], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (4/5, 1/6, [5/6, 6/7, 7/8], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
81. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=5/6, positive finite variance=6/7, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/7, finite prefix of sample means=[6/7, 7/8, 8/9]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(5/6, 1/7, [6/7, 7/8, 8/9], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (5/6, 1/7, [6/7, 7/8, 8/9], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
82. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=6/7, positive finite variance=7/8, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/8, finite prefix of sample means=[7/8, 8/9, 9/10]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(6/7, 1/8, [7/8, 8/9, 9/10], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (6/7, 1/8, [7/8, 8/9, 9/10], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
83. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=7/8, positive finite variance=8/9, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/9, finite prefix of sample means=[8/9, 9/10, 10/11]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(7/8, 1/9, [8/9, 9/10, 10/11], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (7/8, 1/9, [8/9, 9/10, 10/11], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
84. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=8/9, positive finite variance=9/10, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/10, finite prefix of sample means=[9/10, 10/11, 11/12]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(8/9, 1/10, [9/10, 10/11, 11/12], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (8/9, 1/10, [9/10, 10/11, 11/12], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
85. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=9/10, positive finite variance=10/11, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/11, finite prefix of sample means=[10/11, 11/12, 12/13]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(9/10, 1/11, [10/11, 11/12, 12/13], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (9/10, 1/11, [10/11, 11/12, 12/13], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
86. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=10/11, positive finite variance=11/12, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/12, finite prefix of sample means=[11/12, 12/13, 13/14]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(10/11, 1/12, [11/12, 12/13, 13/14], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (10/11, 1/12, [11/12, 12/13, 13/14], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
87. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=11/12, positive finite variance=12/13, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/13, finite prefix of sample means=[12/13, 13/14, 14/15]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(11/12, 1/13, [12/13, 13/14, 14/15], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (11/12, 1/13, [12/13, 13/14, 14/15], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
88. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=12/13, positive finite variance=13/14, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/14, finite prefix of sample means=[13/14, 14/15, 15/16]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(12/13, 1/14, [13/14, 14/15, 15/16], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (12/13, 1/14, [13/14, 14/15, 15/16], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
89. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=13/14, positive finite variance=14/15, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/15, finite prefix of sample means=[14/15, 15/16, 16/17]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(13/14, 1/15, [14/15, 15/16, 16/17], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (13/14, 1/15, [14/15, 15/16, 16/17], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
90. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=14/15, positive finite variance=15/16, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/16, finite prefix of sample means=[15/16, 16/17, 17/18]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(14/15, 1/16, [15/16, 16/17, 17/18], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (14/15, 1/16, [15/16, 16/17, 17/18], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
91. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=15/16, positive finite variance=16/17, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/17, finite prefix of sample means=[16/17, 17/18, 18/19]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(15/16, 1/17, [16/17, 17/18, 18/19], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (15/16, 1/17, [16/17, 17/18, 18/19], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
92. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=16/17, positive finite variance=17/18, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/18, finite prefix of sample means=[17/18, 18/19, 19/20]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(16/17, 1/18, [17/18, 18/19, 19/20], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (16/17, 1/18, [17/18, 18/19, 19/20], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
93. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=17/18, positive finite variance=18/19, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/19, finite prefix of sample means=[18/19, 19/20, 20/21]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(17/18, 1/19, [18/19, 19/20, 20/21], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (17/18, 1/19, [18/19, 19/20, 20/21], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
94. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=18/19, positive finite variance=19/20, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/20, finite prefix of sample means=[19/20, 20/21, 21/22]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(18/19, 1/20, [19/20, 20/21, 21/22], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (18/19, 1/20, [19/20, 20/21, 21/22], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
95. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=19/20, positive finite variance=20/21, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/21, finite prefix of sample means=[20/21, 21/22, 22/23]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(19/20, 1/21, [20/21, 21/22, 22/23], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (19/20, 1/21, [20/21, 21/22, 22/23], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
96. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=20/21, positive finite variance=21/22, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/22, finite prefix of sample means=[21/22, 22/23, 23/24]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(20/21, 1/22, [21/22, 22/23, 23/24], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (20/21, 1/22, [21/22, 22/23, 23/24], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
97. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=21/22, positive finite variance=22/23, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/23, finite prefix of sample means=[22/23, 23/24, 24/25]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(21/22, 1/23, [22/23, 23/24, 24/25], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (21/22, 1/23, [22/23, 23/24, 24/25], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
98. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=22/23, positive finite variance=23/24, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/24, finite prefix of sample means=[23/24, 24/25, 25/26]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(22/23, 1/24, [23/24, 24/25, 25/26], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (22/23, 1/24, [23/24, 24/25, 25/26], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
99. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=23/24, positive finite variance=24/25, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/25, finite prefix of sample means=[24/25, 25/26, 26/27]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(23/24, 1/25, [24/25, 25/26, 26/27], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (23/24, 1/25, [24/25, 25/26, 26/27], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
100. A Finite Simulation Does Not Certify the Whole IID Sequence
Use declared mean=24/25, positive finite variance=25/26, and iid_certificate=finite_simulation_only. In the symbolic theorem families this marker supplies the premise of common mean/variance and IID for every finite subfamily of the whole sequence; a token or finite simulation is not independent proof of that premise. Independence then gives Var(sample mean_n)=variance/n and Chebyshev controls P(|sample mean_n−mean|≥epsilon). This finite-variance WLLN slice claims convergence in probability, not the strong law, CLT or finite-n equality with a limit. Positive epsilon=1/26, finite prefix of sample means=[25/26, 26/27, 27/28]. The supplied marker is finite_simulation_only, not the symbolic whole-sequence premise. Give (mean, epsilon, prefix_sample_means, sufficient, boundary).
View answer and explanation
(24/25, 1/26, [25/26, 26/27, 27/28], No, finite_simulation_not_IID_certificate_or_WLLN_proof)
Retain the exact observed prefix and No for sufficiency. No finite prefix alone proves IID for arbitrary finite subfamilies, the WLLN tail or a stronger convergence mode. The exact result, retaining the requested order, is (24/25, 1/26, [25/26, 26/27, 27/28], No, finite_simulation_not_IID_certificate_or_WLLN_proof).
Reviewed step-by-step derivation
Weak law of large numbers · qf-mod36-wlln-finite-simulation-boundary · candidate
Canonical question JSONL · English JSONL · Validation report