# 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](https://ocme.evemisslab.com/data/i18n/question-output-protocols.json)



## 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0100/)

Chebyshev inequality · qf-mod39-cheb-markov-square-derivation · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations/validation-report.json)