Exact Tails, Raw Markov Bounds and Nonnegative-Premise Boundaries
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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-20261107-r04-markov-inequality-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/6, 2/6, 3/6] and aligned total values X=[0, 1, 3]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/6, 1/3, 1/2], [0, 1, 3], Yes, 11/6, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/6, 1/3, 1/2], [0, 1, 3], Yes, 11/6, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
2. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/7, 2/7, 4/7] and aligned total values X=[0, 2, 4]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/7, 2/7, 4/7], [0, 2, 4], Yes, 20/7, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/7, 2/7, 4/7], [0, 2, 4], Yes, 20/7, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
3. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/8, 2/8, 5/8] and aligned total values X=[0, 3, 5]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/8, 1/4, 5/8], [0, 3, 5], Yes, 31/8, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/8, 1/4, 5/8], [0, 3, 5], Yes, 31/8, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
4. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/9, 2/9, 6/9] and aligned total values X=[0, 4, 6]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/9, 2/9, 2/3], [0, 4, 6], Yes, 44/9, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/9, 2/9, 2/3], [0, 4, 6], Yes, 44/9, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
5. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/10, 2/10, 7/10] and aligned total values X=[0, 5, 7]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/10, 1/5, 7/10], [0, 5, 7], Yes, 59/10, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/10, 1/5, 7/10], [0, 5, 7], Yes, 59/10, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
6. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/11, 2/11, 8/11] and aligned total values X=[0, 6, 8]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/11, 2/11, 8/11], [0, 6, 8], Yes, 76/11, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/11, 2/11, 8/11], [0, 6, 8], Yes, 76/11, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
7. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/12, 2/12, 9/12] and aligned total values X=[0, 7, 9]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/12, 1/6, 3/4], [0, 7, 9], Yes, 95/12, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/12, 1/6, 3/4], [0, 7, 9], Yes, 95/12, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
8. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/13, 2/13, 10/13] and aligned total values X=[0, 8, 10]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/13, 2/13, 10/13], [0, 8, 10], Yes, 116/13, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/13, 2/13, 10/13], [0, 8, 10], Yes, 116/13, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
9. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/14, 2/14, 11/14] and aligned total values X=[0, 9, 11]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/14, 1/7, 11/14], [0, 9, 11], Yes, 139/14, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/14, 1/7, 11/14], [0, 9, 11], Yes, 139/14, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
10. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/15, 2/15, 12/15] and aligned total values X=[0, 10, 12]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/15, 2/15, 4/5], [0, 10, 12], Yes, 164/15, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/15, 2/15, 4/5], [0, 10, 12], Yes, 164/15, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
11. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/16, 2/16, 13/16] and aligned total values X=[0, 11, 13]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/16, 1/8, 13/16], [0, 11, 13], Yes, 191/16, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/16, 1/8, 13/16], [0, 11, 13], Yes, 191/16, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
12. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/17, 2/17, 14/17] and aligned total values X=[0, 12, 14]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/17, 2/17, 14/17], [0, 12, 14], Yes, 220/17, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/17, 2/17, 14/17], [0, 12, 14], Yes, 220/17, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
13. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/18, 2/18, 15/18] and aligned total values X=[0, 13, 15]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/18, 1/9, 5/6], [0, 13, 15], Yes, 251/18, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/18, 1/9, 5/6], [0, 13, 15], Yes, 251/18, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
14. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/19, 2/19, 16/19] and aligned total values X=[0, 14, 16]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/19, 2/19, 16/19], [0, 14, 16], Yes, 284/19, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/19, 2/19, 16/19], [0, 14, 16], Yes, 284/19, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
15. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/20, 2/20, 17/20] and aligned total values X=[0, 15, 17]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/20, 1/10, 17/20], [0, 15, 17], Yes, 319/20, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/20, 1/10, 17/20], [0, 15, 17], Yes, 319/20, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
16. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/21, 2/21, 18/21] and aligned total values X=[0, 16, 18]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/21, 2/21, 6/7], [0, 16, 18], Yes, 356/21, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/21, 2/21, 6/7], [0, 16, 18], Yes, 356/21, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
17. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/22, 2/22, 19/22] and aligned total values X=[0, 17, 19]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/22, 1/11, 19/22], [0, 17, 19], Yes, 395/22, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/22, 1/11, 19/22], [0, 17, 19], Yes, 395/22, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
18. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/23, 2/23, 20/23] and aligned total values X=[0, 18, 20]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/23, 2/23, 20/23], [0, 18, 20], Yes, 436/23, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/23, 2/23, 20/23], [0, 18, 20], Yes, 436/23, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
19. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/24, 2/24, 21/24] and aligned total values X=[0, 19, 21]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/24, 1/12, 7/8], [0, 19, 21], Yes, 479/24, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/24, 1/12, 7/8], [0, 19, 21], Yes, 479/24, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
20. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/25, 2/25, 22/25] and aligned total values X=[0, 20, 22]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/25, 2/25, 22/25], [0, 20, 22], Yes, 524/25, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/25, 2/25, 22/25], [0, 20, 22], Yes, 524/25, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
21. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/26, 2/26, 23/26] and aligned total values X=[0, 21, 23]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/26, 1/13, 23/26], [0, 21, 23], Yes, 571/26, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/26, 1/13, 23/26], [0, 21, 23], Yes, 571/26, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
22. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/27, 2/27, 24/27] and aligned total values X=[0, 22, 24]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/27, 2/27, 8/9], [0, 22, 24], Yes, 620/27, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/27, 2/27, 8/9], [0, 22, 24], Yes, 620/27, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
23. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/28, 2/28, 25/28] and aligned total values X=[0, 23, 25]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/28, 1/14, 25/28], [0, 23, 25], Yes, 671/28, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/28, 1/14, 25/28], [0, 23, 25], Yes, 671/28, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
24. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/29, 2/29, 26/29] and aligned total values X=[0, 24, 26]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/29, 2/29, 26/29], [0, 24, 26], Yes, 724/29, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/29, 2/29, 26/29], [0, 24, 26], Yes, 724/29, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
25. Keep the Source Pointwise Check Distinct from Almost-Sure Nonnegativity
Use complete normalized finite law with exact masses=[1/30, 2/30, 27/30] and aligned total values X=[0, 25, 27]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Give (masses, values, all_values_nonnegative, expectation, source_Markov_qualification, certificate).
The prototype checks nonnegativity at every table entry; almost-sure nonnegativity only constrains positive-mass points. Null-mass negative-value ambiguities need separate source review.
View answer and explanation
([1/30, 1/15, 9/10], [0, 25, 27], Yes, 779/30, Yes, finite_expectation_and_nonnegative_almost_sure)
Return normalized masses, values, the every-entry nonnegativity flag and exact expectation. The source qualification repeats that pointwise flag, a sufficient condition for almost-sure nonnegativity. Null-mass-only negative ambiguities are rejected by the English adapter for source review, not converted into a false theorem claim. The exact result, retaining the requested order, is ([1/30, 1/15, 9/10], [0, 25, 27], Yes, 779/30, Yes, finite_expectation_and_nonnegative_almost_sure).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-qualification · candidate
26. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/6, 2/6, 3/6] and aligned total values X=[0, 1, 3]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=2. Give (threshold, tail_probability, certificate).
View answer and explanation
(2, 1/2, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (2, 1/2, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
27. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/7, 2/7, 4/7] and aligned total values X=[0, 2, 4]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=3. Give (threshold, tail_probability, certificate).
View answer and explanation
(3, 4/7, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (3, 4/7, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
28. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/8, 2/8, 5/8] and aligned total values X=[0, 3, 5]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=4. Give (threshold, tail_probability, certificate).
View answer and explanation
(4, 5/8, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (4, 5/8, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
29. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/9, 2/9, 6/9] and aligned total values X=[0, 4, 6]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=5. Give (threshold, tail_probability, certificate).
View answer and explanation
(5, 2/3, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (5, 2/3, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
30. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/10, 2/10, 7/10] and aligned total values X=[0, 5, 7]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=6. Give (threshold, tail_probability, certificate).
View answer and explanation
(6, 7/10, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (6, 7/10, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
31. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/11, 2/11, 8/11] and aligned total values X=[0, 6, 8]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=7. Give (threshold, tail_probability, certificate).
View answer and explanation
(7, 8/11, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (7, 8/11, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
32. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/12, 2/12, 9/12] and aligned total values X=[0, 7, 9]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=8. Give (threshold, tail_probability, certificate).
View answer and explanation
(8, 3/4, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (8, 3/4, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
33. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/13, 2/13, 10/13] and aligned total values X=[0, 8, 10]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=9. Give (threshold, tail_probability, certificate).
View answer and explanation
(9, 10/13, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (9, 10/13, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
34. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/14, 2/14, 11/14] and aligned total values X=[0, 9, 11]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=10. Give (threshold, tail_probability, certificate).
View answer and explanation
(10, 11/14, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (10, 11/14, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
35. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/15, 2/15, 12/15] and aligned total values X=[0, 10, 12]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=11. Give (threshold, tail_probability, certificate).
View answer and explanation
(11, 4/5, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (11, 4/5, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
36. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/16, 2/16, 13/16] and aligned total values X=[0, 11, 13]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=12. Give (threshold, tail_probability, certificate).
View answer and explanation
(12, 13/16, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (12, 13/16, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
37. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/17, 2/17, 14/17] and aligned total values X=[0, 12, 14]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=13. Give (threshold, tail_probability, certificate).
View answer and explanation
(13, 14/17, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (13, 14/17, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
38. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/18, 2/18, 15/18] and aligned total values X=[0, 13, 15]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=14. Give (threshold, tail_probability, certificate).
View answer and explanation
(14, 5/6, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (14, 5/6, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
39. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/19, 2/19, 16/19] and aligned total values X=[0, 14, 16]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=15. Give (threshold, tail_probability, certificate).
View answer and explanation
(15, 16/19, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (15, 16/19, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
40. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/20, 2/20, 17/20] and aligned total values X=[0, 15, 17]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=16. Give (threshold, tail_probability, certificate).
View answer and explanation
(16, 17/20, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (16, 17/20, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
41. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/21, 2/21, 18/21] and aligned total values X=[0, 16, 18]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=17. Give (threshold, tail_probability, certificate).
View answer and explanation
(17, 6/7, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (17, 6/7, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
42. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/22, 2/22, 19/22] and aligned total values X=[0, 17, 19]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=18. Give (threshold, tail_probability, certificate).
View answer and explanation
(18, 19/22, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (18, 19/22, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
43. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/23, 2/23, 20/23] and aligned total values X=[0, 18, 20]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=19. Give (threshold, tail_probability, certificate).
View answer and explanation
(19, 20/23, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (19, 20/23, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
44. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/24, 2/24, 21/24] and aligned total values X=[0, 19, 21]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=20. Give (threshold, tail_probability, certificate).
View answer and explanation
(20, 7/8, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (20, 7/8, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
45. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/25, 2/25, 22/25] and aligned total values X=[0, 20, 22]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=21. Give (threshold, tail_probability, certificate).
View answer and explanation
(21, 22/25, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (21, 22/25, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
46. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/26, 2/26, 23/26] and aligned total values X=[0, 21, 23]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=22. Give (threshold, tail_probability, certificate).
View answer and explanation
(22, 23/26, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (22, 23/26, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
47. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/27, 2/27, 24/27] and aligned total values X=[0, 22, 24]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=23. Give (threshold, tail_probability, certificate).
View answer and explanation
(23, 8/9, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (23, 8/9, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
48. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/28, 2/28, 25/28] and aligned total values X=[0, 23, 25]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=24. Give (threshold, tail_probability, certificate).
View answer and explanation
(24, 25/28, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (24, 25/28, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
49. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/29, 2/29, 26/29] and aligned total values X=[0, 24, 26]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=25. Give (threshold, tail_probability, certificate).
View answer and explanation
(25, 26/29, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (25, 26/29, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
50. The Exact Tail Is a Sum over the Full Threshold Event
Use complete normalized finite law with exact masses=[1/30, 2/30, 27/30] and aligned total values X=[0, 25, 27]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=26. Give (threshold, tail_probability, certificate).
View answer and explanation
(26, 9/10, exact_complete_finite_tail)
Add exactly the masses at outcomes with X≥threshold, including equality. This event probability can be computed for a signed variable as well; calculating it alone does not assert Markov applicability. The exact result, retaining the requested order, is (26, 9/10, exact_complete_finite_tail).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-tail-probability · candidate
51. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/6, 2/6, 3/6] and aligned total values X=[0, 1, 3]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=2>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(1/2, 11/6, 11/12, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (1/2, 11/6, 11/12, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
52. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/7, 2/7, 4/7] and aligned total values X=[0, 2, 4]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=3>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(4/7, 20/7, 20/21, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (4/7, 20/7, 20/21, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
53. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/8, 2/8, 5/8] and aligned total values X=[0, 3, 5]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=4>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(5/8, 31/8, 31/32, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (5/8, 31/8, 31/32, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
54. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/9, 2/9, 6/9] and aligned total values X=[0, 4, 6]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=5>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(2/3, 44/9, 44/45, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (2/3, 44/9, 44/45, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
55. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/10, 2/10, 7/10] and aligned total values X=[0, 5, 7]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=6>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(7/10, 59/10, 59/60, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (7/10, 59/10, 59/60, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
56. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/11, 2/11, 8/11] and aligned total values X=[0, 6, 8]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=7>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(8/11, 76/11, 76/77, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (8/11, 76/11, 76/77, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
57. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/12, 2/12, 9/12] and aligned total values X=[0, 7, 9]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=8>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(3/4, 95/12, 95/96, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (3/4, 95/12, 95/96, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
58. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/13, 2/13, 10/13] and aligned total values X=[0, 8, 10]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=9>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(10/13, 116/13, 116/117, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (10/13, 116/13, 116/117, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
59. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/14, 2/14, 11/14] and aligned total values X=[0, 9, 11]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=10>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(11/14, 139/14, 139/140, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (11/14, 139/14, 139/140, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
60. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/15, 2/15, 12/15] and aligned total values X=[0, 10, 12]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=11>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(4/5, 164/15, 164/165, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (4/5, 164/15, 164/165, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
61. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/16, 2/16, 13/16] and aligned total values X=[0, 11, 13]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=12>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(13/16, 191/16, 191/192, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (13/16, 191/16, 191/192, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
62. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/17, 2/17, 14/17] and aligned total values X=[0, 12, 14]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=13>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(14/17, 220/17, 220/221, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (14/17, 220/17, 220/221, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
63. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/18, 2/18, 15/18] and aligned total values X=[0, 13, 15]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=14>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(5/6, 251/18, 251/252, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (5/6, 251/18, 251/252, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
64. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/19, 2/19, 16/19] and aligned total values X=[0, 14, 16]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=15>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(16/19, 284/19, 284/285, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (16/19, 284/19, 284/285, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
65. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/20, 2/20, 17/20] and aligned total values X=[0, 15, 17]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=16>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(17/20, 319/20, 319/320, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (17/20, 319/20, 319/320, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
66. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/21, 2/21, 18/21] and aligned total values X=[0, 16, 18]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=17>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(6/7, 356/21, 356/357, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (6/7, 356/21, 356/357, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
67. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/22, 2/22, 19/22] and aligned total values X=[0, 17, 19]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=18>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(19/22, 395/22, 395/396, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (19/22, 395/22, 395/396, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
68. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/23, 2/23, 20/23] and aligned total values X=[0, 18, 20]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=19>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(20/23, 436/23, 436/437, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (20/23, 436/23, 436/437, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
69. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/24, 2/24, 21/24] and aligned total values X=[0, 19, 21]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=20>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(7/8, 479/24, 479/480, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (7/8, 479/24, 479/480, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
70. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/25, 2/25, 22/25] and aligned total values X=[0, 20, 22]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=21>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(22/25, 524/25, 524/525, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (22/25, 524/25, 524/525, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
71. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/26, 2/26, 23/26] and aligned total values X=[0, 21, 23]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=22>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(23/26, 571/26, 571/572, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (23/26, 571/26, 571/572, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
72. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/27, 2/27, 24/27] and aligned total values X=[0, 22, 24]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=23>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(8/9, 620/27, 620/621, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (8/9, 620/27, 620/621, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
73. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/28, 2/28, 25/28] and aligned total values X=[0, 23, 25]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=24>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(25/28, 671/28, 671/672, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (25/28, 671/28, 671/672, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
74. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/29, 2/29, 26/29] and aligned total values X=[0, 24, 26]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=25>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(26/29, 724/29, 724/725, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (26/29, 724/29, 724/725, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
75. A Markov Upper Bound Is Not the Tail Probability
Use complete normalized finite law with exact masses=[1/30, 2/30, 27/30] and aligned total values X=[0, 25, 27]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=26>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).
raw_upper_bound may exceed1; it is not the actual tail probability and need not equal it.
View answer and explanation
(9/10, 779/30, 779/780, Yes, upper_bound_not_tail_equality)
Compute the actual tail, E[X] and E[X]/threshold separately, then check tail≤bound. The upper bound can exceed1 or be strictly larger than the tail; neither makes it an invalid probability value because it is not the probability itself. The exact result, retaining the requested order, is (9/10, 779/30, 779/780, Yes, upper_bound_not_tail_equality).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-bound-comparison · candidate
76. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-1, 3]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=2, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-1, 3], No, 1/2, 1/2, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-1, 3], No, 1/2, 1/2, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
77. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-2, 4]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=3, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-2, 4], No, 1/2, 1/3, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-2, 4], No, 1/2, 1/3, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
78. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-3, 5]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=4, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-3, 5], No, 1/2, 1/4, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-3, 5], No, 1/2, 1/4, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
79. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-4, 6]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=5, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-4, 6], No, 1/2, 1/5, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-4, 6], No, 1/2, 1/5, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
80. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-5, 7]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=6, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-5, 7], No, 1/2, 1/6, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-5, 7], No, 1/2, 1/6, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
81. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-6, 8]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=7, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-6, 8], No, 1/2, 1/7, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-6, 8], No, 1/2, 1/7, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
82. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-7, 9]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=8, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-7, 9], No, 1/2, 1/8, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-7, 9], No, 1/2, 1/8, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
83. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-8, 10]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=9, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-8, 10], No, 1/2, 1/9, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-8, 10], No, 1/2, 1/9, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
84. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-9, 11]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=10, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-9, 11], No, 1/2, 1/10, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-9, 11], No, 1/2, 1/10, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
85. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-10, 12]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=11, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-10, 12], No, 1/2, 1/11, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-10, 12], No, 1/2, 1/11, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
86. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-11, 13]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=12, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-11, 13], No, 1/2, 1/12, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-11, 13], No, 1/2, 1/12, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
87. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-12, 14]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=13, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-12, 14], No, 1/2, 1/13, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-12, 14], No, 1/2, 1/13, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
88. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-13, 15]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=14, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-13, 15], No, 1/2, 1/14, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-13, 15], No, 1/2, 1/14, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
89. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-14, 16]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=15, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-14, 16], No, 1/2, 1/15, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-14, 16], No, 1/2, 1/15, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
90. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-15, 17]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=16, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-15, 17], No, 1/2, 1/16, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-15, 17], No, 1/2, 1/16, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
91. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-16, 18]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=17, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-16, 18], No, 1/2, 1/17, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-16, 18], No, 1/2, 1/17, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
92. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-17, 19]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=18, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-17, 19], No, 1/2, 1/18, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-17, 19], No, 1/2, 1/18, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
93. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-18, 20]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=19, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-18, 20], No, 1/2, 1/19, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-18, 20], No, 1/2, 1/19, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
94. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-19, 21]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=20, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-19, 21], No, 1/2, 1/20, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-19, 21], No, 1/2, 1/20, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
95. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-20, 22]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=21, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-20, 22], No, 1/2, 1/21, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-20, 22], No, 1/2, 1/21, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
96. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-21, 23]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=22, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-21, 23], No, 1/2, 1/22, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-21, 23], No, 1/2, 1/22, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
97. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-22, 24]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=23, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-22, 24], No, 1/2, 1/23, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-22, 24], No, 1/2, 1/23, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
98. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-23, 25]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=24, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-23, 25], No, 1/2, 1/24, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-23, 25], No, 1/2, 1/24, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
99. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-24, 26]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=25, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-24, 26], No, 1/2, 1/25, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-24, 26], No, 1/2, 1/25, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
100. A Failed Nonnegativity Premise Does Not Force This Particular Inequality to Fail
Use complete normalized finite law with exact masses=[1/2, 1/2] and aligned total values X=[-25, 27]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. Positive threshold=26, with a negative value at positive mass in this boundary case. Give (values, pointwise_Markov_applicable, tail_probability, raw_expectation_over_threshold, boundary).
The inapplicability marker does not say this particular inequality must fail or reverse; without nonnegativity the prototype supplies no general Markov guarantee.
View answer and explanation
([-25, 27], No, 1/2, 1/26, Markov_not_applicable_to_signed_variable_without_transformation)
Report the value table, No for source qualification, the exact tail and the raw quotient E[X]/threshold. Without the nonnegative premise Markov provides no general guarantee; the numeric inequality may fail or coincidentally hold. Do not turn inapplicability into a proved reverse inequality. The exact result, retaining the requested order, is ([-25, 27], No, 1/2, 1/26, Markov_not_applicable_to_signed_variable_without_transformation).
Reviewed step-by-step derivation
Markov inequality · qf-mod38-markov-negative-variable-boundary · candidate
Canonical question JSONL · English JSONL · Validation report