Exact Deviation Tails, Raw Chebyshev Bounds and Squared-Event Derivation
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This question bank has corresponding bilingual step-by-step derivations. Source candidate and English ai_candidate states are retained. AI review, finite computation and Lean formal proof are labelled separately.
batch-20261014-advance-20261108-r04-chebyshev-inequality-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. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/6, 2/6, 3/6] and aligned values X=[0, 1, 4]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/6, 1/3, 1/2], [0, 1, 4], 7/3, 26/9, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/6, 1/3, 1/2], [0, 1, 4], 7/3, 26/9, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
2. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/7, 2/7, 4/7] and aligned values X=[0, 2, 5]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/7, 2/7, 4/7], [0, 2, 5], 24/7, 180/49, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/7, 2/7, 4/7], [0, 2, 5], 24/7, 180/49, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
3. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/8, 2/8, 5/8] and aligned values X=[0, 3, 6]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/8, 1/4, 5/8], [0, 3, 6], 9/2, 9/2, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/8, 1/4, 5/8], [0, 3, 6], 9/2, 9/2, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
4. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/9, 2/9, 6/9] and aligned values X=[0, 4, 7]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/9, 2/9, 2/3], [0, 4, 7], 50/9, 434/81, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/9, 2/9, 2/3], [0, 4, 7], 50/9, 434/81, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
5. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/10, 2/10, 7/10] and aligned values X=[0, 5, 8]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/10, 1/5, 7/10], [0, 5, 8], 33/5, 156/25, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/10, 1/5, 7/10], [0, 5, 8], 33/5, 156/25, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
6. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/11, 2/11, 8/11], [0, 6, 9], 84/11, 864/121, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/11, 2/11, 8/11], [0, 6, 9], 84/11, 864/121, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
7. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/12, 2/12, 9/12] and aligned values X=[0, 7, 10]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/12, 1/6, 3/4], [0, 7, 10], 26/3, 145/18, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/12, 1/6, 3/4], [0, 7, 10], 26/3, 145/18, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
8. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/13, 2/13, 10/13] and aligned values X=[0, 8, 11]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/13, 2/13, 10/13], [0, 8, 11], 126/13, 1518/169, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/13, 2/13, 10/13], [0, 8, 11], 126/13, 1518/169, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
9. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/14, 2/14, 11/14] and aligned values X=[0, 9, 12]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/14, 1/7, 11/14], [0, 9, 12], 75/7, 486/49, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/14, 1/7, 11/14], [0, 9, 12], 75/7, 486/49, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
10. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/15, 2/15, 12/15] and aligned values X=[0, 10, 13]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/15, 2/15, 4/5], [0, 10, 13], 176/15, 2444/225, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/15, 2/15, 4/5], [0, 10, 13], 176/15, 2444/225, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
11. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/16, 2/16, 13/16] and aligned values X=[0, 11, 14]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/16, 1/8, 13/16], [0, 11, 14], 51/4, 189/16, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/16, 1/8, 13/16], [0, 11, 14], 51/4, 189/16, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
12. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/17, 2/17, 14/17] and aligned values X=[0, 12, 15]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/17, 2/17, 14/17], [0, 12, 15], 234/17, 3690/289, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/17, 2/17, 14/17], [0, 12, 15], 234/17, 3690/289, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
13. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/18, 2/18, 15/18] and aligned values X=[0, 13, 16]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/18, 1/9, 5/6], [0, 13, 16], 133/9, 1112/81, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/18, 1/9, 5/6], [0, 13, 16], 133/9, 1112/81, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
14. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/19, 2/19, 16/19] and aligned values X=[0, 14, 17]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/19, 2/19, 16/19], [0, 14, 17], 300/19, 5304/361, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/19, 2/19, 16/19], [0, 14, 17], 300/19, 5304/361, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
15. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/20, 2/20, 17/20] and aligned values X=[0, 15, 18]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/20, 1/10, 17/20], [0, 15, 18], 84/5, 783/50, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/20, 1/10, 17/20], [0, 15, 18], 84/5, 783/50, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
16. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/21, 2/21, 18/21] and aligned values X=[0, 16, 19]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/21, 2/21, 6/7], [0, 16, 19], 374/21, 7334/441, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/21, 2/21, 6/7], [0, 16, 19], 374/21, 7334/441, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
17. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/22, 2/22, 19/22] and aligned values X=[0, 17, 20]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/22, 1/11, 19/22], [0, 17, 20], 207/11, 2130/121, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/22, 1/11, 19/22], [0, 17, 20], 207/11, 2130/121, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
18. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/23, 2/23, 20/23] and aligned values X=[0, 18, 21]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/23, 2/23, 20/23], [0, 18, 21], 456/23, 9828/529, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/23, 2/23, 20/23], [0, 18, 21], 456/23, 9828/529, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
19. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/24, 2/24, 21/24] and aligned values X=[0, 19, 22]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/24, 1/12, 7/8], [0, 19, 22], 125/6, 176/9, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/24, 1/12, 7/8], [0, 19, 22], 125/6, 176/9, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
20. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/25, 2/25, 22/25] and aligned values X=[0, 20, 23]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/25, 2/25, 22/25], [0, 20, 23], 546/25, 12834/625, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/25, 2/25, 22/25], [0, 20, 23], 546/25, 12834/625, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
21. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/26, 2/26, 23/26] and aligned values X=[0, 21, 24]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/26, 1/13, 23/26], [0, 21, 24], 297/13, 3636/169, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/26, 1/13, 23/26], [0, 21, 24], 297/13, 3636/169, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
22. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/27, 2/27, 24/27] and aligned values X=[0, 22, 25]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/27, 2/27, 8/9], [0, 22, 25], 644/27, 16400/729, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/27, 2/27, 8/9], [0, 22, 25], 644/27, 16400/729, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
23. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/28, 2/28, 25/28] and aligned values X=[0, 23, 26]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/28, 1/14, 25/28], [0, 23, 26], 174/7, 2301/98, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/28, 1/14, 25/28], [0, 23, 26], 174/7, 2301/98, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
24. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/29, 2/29, 26/29] and aligned values X=[0, 24, 27]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/29, 2/29, 26/29], [0, 24, 27], 750/29, 20574/841, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/29, 2/29, 26/29], [0, 24, 27], 750/29, 20574/841, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
25. Finite Mean and Variance Do Not Need a Nonnegative Original Variable
Use complete normalized finite law masses=[1/30, 2/30, 27/30] and aligned values X=[0, 25, 28]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Give (masses, values, mean, variance, finite_moment_certificate, certificate).
View answer and explanation
([1/30, 1/15, 9/10], [0, 25, 28], 403/15, 5726/225, Yes, finite_mean_and_variance)
Compute exact normalized masses, values, mean and variance. Signed X is allowed; the nonnegative variable used in the derivation is the squared centered deviation. These finite moments are not assumptions of independence. The exact result, retaining the requested order, is ([1/30, 1/15, 9/10], [0, 25, 28], 403/15, 5726/225, Yes, finite_mean_and_variance).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-qualification · candidate
26. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/6, 2/6, 3/6] and aligned values X=[0, 1, 4]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(7/3, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (7/3, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
27. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/7, 2/7, 4/7] and aligned values X=[0, 2, 5]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(24/7, 2, 1/7, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (24/7, 2, 1/7, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
28. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/8, 2/8, 5/8] and aligned values X=[0, 3, 6]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(9/2, 3, 1/8, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (9/2, 3, 1/8, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
29. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/9, 2/9, 6/9] and aligned values X=[0, 4, 7]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(50/9, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (50/9, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
30. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/10, 2/10, 7/10] and aligned values X=[0, 5, 8]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(33/5, 2, 1/10, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (33/5, 2, 1/10, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
31. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(84/11, 3, 1/11, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (84/11, 3, 1/11, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
32. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/12, 2/12, 9/12] and aligned values X=[0, 7, 10]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(26/3, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (26/3, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
33. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/13, 2/13, 10/13] and aligned values X=[0, 8, 11]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(126/13, 2, 1/13, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (126/13, 2, 1/13, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
34. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/14, 2/14, 11/14] and aligned values X=[0, 9, 12]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(75/7, 3, 1/14, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (75/7, 3, 1/14, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
35. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/15, 2/15, 12/15] and aligned values X=[0, 10, 13]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(176/15, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (176/15, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
36. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/16, 2/16, 13/16] and aligned values X=[0, 11, 14]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(51/4, 2, 1/16, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (51/4, 2, 1/16, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
37. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/17, 2/17, 14/17] and aligned values X=[0, 12, 15]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(234/17, 3, 1/17, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (234/17, 3, 1/17, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
38. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/18, 2/18, 15/18] and aligned values X=[0, 13, 16]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(133/9, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (133/9, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
39. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/19, 2/19, 16/19] and aligned values X=[0, 14, 17]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(300/19, 2, 1/19, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (300/19, 2, 1/19, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
40. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/20, 2/20, 17/20] and aligned values X=[0, 15, 18]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(84/5, 3, 1/20, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (84/5, 3, 1/20, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
41. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/21, 2/21, 18/21] and aligned values X=[0, 16, 19]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(374/21, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (374/21, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
42. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/22, 2/22, 19/22] and aligned values X=[0, 17, 20]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(207/11, 2, 1/22, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (207/11, 2, 1/22, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
43. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/23, 2/23, 20/23] and aligned values X=[0, 18, 21]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(456/23, 3, 1/23, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (456/23, 3, 1/23, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
44. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/24, 2/24, 21/24] and aligned values X=[0, 19, 22]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(125/6, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (125/6, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
45. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/25, 2/25, 22/25] and aligned values X=[0, 20, 23]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(546/25, 2, 1/25, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (546/25, 2, 1/25, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
46. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/26, 2/26, 23/26] and aligned values X=[0, 21, 24]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(297/13, 3, 1/26, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (297/13, 3, 1/26, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
47. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/27, 2/27, 24/27] and aligned values X=[0, 22, 25]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(644/27, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (644/27, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
48. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/28, 2/28, 25/28] and aligned values X=[0, 23, 26]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(174/7, 2, 1/28, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (174/7, 2, 1/28, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
49. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/29, 2/29, 26/29] and aligned values X=[0, 24, 27]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(750/29, 3, 1/29, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (750/29, 3, 1/29, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
50. The Absolute-Deviation Event Includes Equality at Epsilon
Use complete normalized finite law masses=[1/30, 2/30, 27/30] and aligned values X=[0, 25, 28]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (mean, epsilon, tail_probability, certificate).
View answer and explanation
(403/15, 1, 1, exact_absolute_deviation_tail)
Sum the complete law over |X−mean|≥epsilon. Retain the exact mean, epsilon and tail; the threshold event is non-strict even though epsilon itself must be positive. The exact result, retaining the requested order, is (403/15, 1, 1, exact_absolute_deviation_tail).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-tail-probability · candidate
51. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/6, 2/6, 3/6] and aligned values X=[0, 1, 4]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 26/9, 26/9, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 26/9, 26/9, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
52. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/7, 2/7, 4/7] and aligned values X=[0, 2, 5]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/7, 180/49, 45/49, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/7, 180/49, 45/49, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
53. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/8, 2/8, 5/8] and aligned values X=[0, 3, 6]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/8, 9/2, 1/2, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/8, 9/2, 1/2, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
54. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/9, 2/9, 6/9] and aligned values X=[0, 4, 7]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 434/81, 434/81, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 434/81, 434/81, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
55. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/10, 2/10, 7/10] and aligned values X=[0, 5, 8]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/10, 156/25, 39/25, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/10, 156/25, 39/25, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
56. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/11, 864/121, 96/121, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/11, 864/121, 96/121, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
57. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/12, 2/12, 9/12] and aligned values X=[0, 7, 10]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 145/18, 145/18, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 145/18, 145/18, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
58. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/13, 2/13, 10/13] and aligned values X=[0, 8, 11]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/13, 1518/169, 759/338, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/13, 1518/169, 759/338, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
59. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/14, 2/14, 11/14] and aligned values X=[0, 9, 12]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/14, 486/49, 54/49, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/14, 486/49, 54/49, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
60. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/15, 2/15, 12/15] and aligned values X=[0, 10, 13]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 2444/225, 2444/225, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 2444/225, 2444/225, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
61. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/16, 2/16, 13/16] and aligned values X=[0, 11, 14]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/16, 189/16, 189/64, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/16, 189/16, 189/64, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
62. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/17, 2/17, 14/17] and aligned values X=[0, 12, 15]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/17, 3690/289, 410/289, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/17, 3690/289, 410/289, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
63. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/18, 2/18, 15/18] and aligned values X=[0, 13, 16]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 1112/81, 1112/81, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 1112/81, 1112/81, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
64. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/19, 2/19, 16/19] and aligned values X=[0, 14, 17]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/19, 5304/361, 1326/361, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/19, 5304/361, 1326/361, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
65. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/20, 2/20, 17/20] and aligned values X=[0, 15, 18]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/20, 783/50, 87/50, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/20, 783/50, 87/50, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
66. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/21, 2/21, 18/21] and aligned values X=[0, 16, 19]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 7334/441, 7334/441, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 7334/441, 7334/441, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
67. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/22, 2/22, 19/22] and aligned values X=[0, 17, 20]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/22, 2130/121, 1065/242, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/22, 2130/121, 1065/242, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
68. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/23, 2/23, 20/23] and aligned values X=[0, 18, 21]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/23, 9828/529, 1092/529, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/23, 9828/529, 1092/529, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
69. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/24, 2/24, 21/24] and aligned values X=[0, 19, 22]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 176/9, 176/9, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 176/9, 176/9, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
70. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/25, 2/25, 22/25] and aligned values X=[0, 20, 23]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/25, 12834/625, 6417/1250, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/25, 12834/625, 6417/1250, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
71. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/26, 2/26, 23/26] and aligned values X=[0, 21, 24]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/26, 3636/169, 404/169, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/26, 3636/169, 404/169, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
72. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/27, 2/27, 24/27] and aligned values X=[0, 22, 25]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 16400/729, 16400/729, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 16400/729, 16400/729, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
73. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/28, 2/28, 25/28] and aligned values X=[0, 23, 26]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/28, 2301/98, 2301/392, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/28, 2301/98, 2301/392, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
74. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/29, 2/29, 26/29] and aligned values X=[0, 24, 27]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1/29, 20574/841, 2286/841, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1/29, 20574/841, 2286/841, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
75. Chebyshev Gives a Raw Upper Bound, Not Tail Equality or Independence
Use complete normalized finite law masses=[1/30, 2/30, 27/30] and aligned values X=[0, 25, 28]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Retain the raw bound without clipping it to1. Give (tail_probability, variance, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1 and need not equal the tail. Chebyshev needs no independence, and zero variance constrains values only almost surely under the same law.
View answer and explanation
(1, 5726/225, 5726/225, Yes, upper_bound_not_tail_equality_and_no_independence_needed)
Compute the actual tail, variance and variance/epsilon^2 separately. The upper bound may exceed1. With variance0, values are constant almost surely under this law, not necessarily at zero-mass table entries. The exact result, retaining the requested order, is (1, 5726/225, 5726/225, Yes, upper_bound_not_tail_equality_and_no_independence_needed).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-bound-comparison · candidate
76. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/6, 2/6, 3/6] and aligned values X=[0, 1, 4]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([49/9, 16/9, 25/9], 1, 1, 1, Yes, 26/9, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([49/9, 16/9, 25/9], 1, 1, 1, Yes, 26/9, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
77. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/7, 2/7, 4/7] and aligned values X=[0, 2, 5]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([576/49, 100/49, 121/49], 4, 1/7, 1/7, Yes, 45/49, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([576/49, 100/49, 121/49], 4, 1/7, 1/7, Yes, 45/49, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
78. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/8, 2/8, 5/8] and aligned values X=[0, 3, 6]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([81/4, 9/4, 9/4], 9, 1/8, 1/8, Yes, 1/2, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([81/4, 9/4, 9/4], 9, 1/8, 1/8, Yes, 1/2, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
79. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/9, 2/9, 6/9] and aligned values X=[0, 4, 7]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([2500/81, 196/81, 169/81], 1, 1, 1, Yes, 434/81, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([2500/81, 196/81, 169/81], 1, 1, 1, Yes, 434/81, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
80. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/10, 2/10, 7/10] and aligned values X=[0, 5, 8]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([1089/25, 64/25, 49/25], 4, 1/10, 1/10, Yes, 39/25, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([1089/25, 64/25, 49/25], 4, 1/10, 1/10, Yes, 39/25, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
81. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([7056/121, 324/121, 225/121], 9, 1/11, 1/11, Yes, 96/121, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([7056/121, 324/121, 225/121], 9, 1/11, 1/11, Yes, 96/121, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
82. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/12, 2/12, 9/12] and aligned values X=[0, 7, 10]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([676/9, 25/9, 16/9], 1, 1, 1, Yes, 145/18, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([676/9, 25/9, 16/9], 1, 1, 1, Yes, 145/18, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
83. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/13, 2/13, 10/13] and aligned values X=[0, 8, 11]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([15876/169, 484/169, 289/169], 4, 1/13, 1/13, Yes, 759/338, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([15876/169, 484/169, 289/169], 4, 1/13, 1/13, Yes, 759/338, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
84. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/14, 2/14, 11/14] and aligned values X=[0, 9, 12]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([5625/49, 144/49, 81/49], 9, 1/14, 1/14, Yes, 54/49, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([5625/49, 144/49, 81/49], 9, 1/14, 1/14, Yes, 54/49, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
85. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/15, 2/15, 12/15] and aligned values X=[0, 10, 13]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([30976/225, 676/225, 361/225], 1, 1, 1, Yes, 2444/225, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([30976/225, 676/225, 361/225], 1, 1, 1, Yes, 2444/225, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
86. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/16, 2/16, 13/16] and aligned values X=[0, 11, 14]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([2601/16, 49/16, 25/16], 4, 1/16, 1/16, Yes, 189/64, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([2601/16, 49/16, 25/16], 4, 1/16, 1/16, Yes, 189/64, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
87. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/17, 2/17, 14/17] and aligned values X=[0, 12, 15]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([54756/289, 900/289, 441/289], 9, 1/17, 1/17, Yes, 410/289, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([54756/289, 900/289, 441/289], 9, 1/17, 1/17, Yes, 410/289, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
88. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/18, 2/18, 15/18] and aligned values X=[0, 13, 16]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([17689/81, 256/81, 121/81], 1, 1, 1, Yes, 1112/81, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([17689/81, 256/81, 121/81], 1, 1, 1, Yes, 1112/81, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
89. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/19, 2/19, 16/19] and aligned values X=[0, 14, 17]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([90000/361, 1156/361, 529/361], 4, 1/19, 1/19, Yes, 1326/361, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([90000/361, 1156/361, 529/361], 4, 1/19, 1/19, Yes, 1326/361, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
90. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/20, 2/20, 17/20] and aligned values X=[0, 15, 18]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([7056/25, 81/25, 36/25], 9, 1/20, 1/20, Yes, 87/50, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([7056/25, 81/25, 36/25], 9, 1/20, 1/20, Yes, 87/50, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
91. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/21, 2/21, 18/21] and aligned values X=[0, 16, 19]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([139876/441, 1444/441, 625/441], 1, 1, 1, Yes, 7334/441, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([139876/441, 1444/441, 625/441], 1, 1, 1, Yes, 7334/441, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
92. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/22, 2/22, 19/22] and aligned values X=[0, 17, 20]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([42849/121, 400/121, 169/121], 4, 1/22, 1/22, Yes, 1065/242, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([42849/121, 400/121, 169/121], 4, 1/22, 1/22, Yes, 1065/242, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
93. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/23, 2/23, 20/23] and aligned values X=[0, 18, 21]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([207936/529, 1764/529, 729/529], 9, 1/23, 1/23, Yes, 1092/529, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([207936/529, 1764/529, 729/529], 9, 1/23, 1/23, Yes, 1092/529, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
94. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/24, 2/24, 21/24] and aligned values X=[0, 19, 22]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([15625/36, 121/36, 49/36], 1, 1, 1, Yes, 176/9, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([15625/36, 121/36, 49/36], 1, 1, 1, Yes, 176/9, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
95. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/25, 2/25, 22/25] and aligned values X=[0, 20, 23]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([298116/625, 2116/625, 841/625], 4, 1/25, 1/25, Yes, 6417/1250, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([298116/625, 2116/625, 841/625], 4, 1/25, 1/25, Yes, 6417/1250, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
96. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/26, 2/26, 23/26] and aligned values X=[0, 21, 24]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([88209/169, 576/169, 225/169], 9, 1/26, 1/26, Yes, 404/169, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([88209/169, 576/169, 225/169], 9, 1/26, 1/26, Yes, 404/169, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
97. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/27, 2/27, 24/27] and aligned values X=[0, 22, 25]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([414736/729, 2500/729, 961/729], 1, 1, 1, Yes, 16400/729, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([414736/729, 2500/729, 961/729], 1, 1, 1, Yes, 16400/729, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
98. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/28, 2/28, 25/28] and aligned values X=[0, 23, 26]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([30276/49, 169/49, 64/49], 4, 1/28, 1/28, Yes, 2301/392, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([30276/49, 169/49, 64/49], 4, 1/28, 1/28, Yes, 2301/392, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
99. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/29, 2/29, 26/29] and aligned values X=[0, 24, 27]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=3/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([562500/841, 2916/841, 1089/841], 9, 1/29, 1/29, Yes, 2286/841, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([562500/841, 2916/841, 1089/841], 9, 1/29, 1/29, Yes, 2286/841, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
100. Apply Markov to the Nonnegative Squared Deviation at Epsilon Squared
Use complete normalized finite law masses=[1/30, 2/30, 27/30] and aligned values X=[0, 25, 28]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=1/1. Give (squared_deviations, epsilon_squared, squared_tail, absolute_tail, same_event, raw_upper_bound, certificate).
View answer and explanation
([162409/225, 784/225, 289/225], 1, 1, 1, Yes, 5726/225, Markov_applied_to_nonnegative_squared_deviation)
Compute every (X−mean)^2, use threshold epsilon^2, and compare the squared-event probability with the absolute-deviation tail. They are the same event. The final raw bound is variance/epsilon^2, not an event-probability equality claim. The exact result, retaining the requested order, is ([162409/225, 784/225, 289/225], 1, 1, 1, Yes, 5726/225, Markov_applied_to_nonnegative_squared_deviation).
Reviewed step-by-step derivation
Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate
Canonical question JSONL · English JSONL · Validation report