Exact Finite PMFs, Events, Expectations and Pushforwards
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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-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations · 2026-10-13 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([0, 1, 3], [1/4, 1/4, 1/4], Yes, 3/4, No, No)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/4], Yes, 3/4, No, No).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
2. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([1, 2, 4], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
3. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([2, 3, 5], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
4. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([3, 4, 6], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
5. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([4, 5, 7], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
6. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([5, 6, 8], [1/4, 1/4, 1/4], Yes, 3/4, No, No)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/4], Yes, 3/4, No, No).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
7. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([6, 7, 9], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
8. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([7, 8, 10], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
9. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([8, 9, 11], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
10. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([9, 10, 12], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
11. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([10, 11, 13], [1/4, 1/4, 1/4], Yes, 3/4, No, No)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/4], Yes, 3/4, No, No).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
12. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([11, 12, 14], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
13. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([12, 13, 15], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
14. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([13, 14, 16], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
15. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([14, 15, 17], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
16. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([15, 16, 18], [1/4, 1/4, 1/4], Yes, 3/4, No, No)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/4], Yes, 3/4, No, No).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
17. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([16, 17, 19], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
18. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([17, 18, 20], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
19. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([18, 19, 21], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
20. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([19, 20, 22], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
21. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/4]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([20, 21, 23], [1/4, 1/4, 1/4], Yes, 3/4, No, No)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/4], Yes, 3/4, No, No).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
22. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([21, 22, 24], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
23. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([22, 23, 25], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
24. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([23, 24, 26], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
25. Exact Normalization and Nonnegativity before PMF Use
Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. This qualification family may supply negative or unnormalized masses to test rejection. Give (original outcomes,normalized rational masses,all masses nonnegative?,exact total,total equals 1?,qualified PMF?).
View answer and explanation
([24, 25, 27], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes)
Check both nonnegativity and exact total. Having entries no larger than 1 does not by itself establish a PMF. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], Yes, 1, Yes, Yes).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pmf-qualification · candidate
26. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[0, 3] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([0, 1, 3], [1/4, 1/4, 1/2], [0, 3], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/2], [0, 3], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
27. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[1, 4] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([1, 2, 4], [1/4, 1/4, 1/2], [1, 4], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], [1, 4], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
28. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[2, 5] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([2, 3, 5], [1/4, 1/4, 1/2], [2, 5], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], [2, 5], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
29. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[3, 6] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([3, 4, 6], [1/4, 1/4, 1/2], [3, 6], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], [3, 6], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
30. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[4, 7] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([4, 5, 7], [1/4, 1/4, 1/2], [4, 7], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], [4, 7], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
31. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[5, 8] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([5, 6, 8], [1/4, 1/4, 1/2], [5, 8], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/2], [5, 8], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
32. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[6, 9] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([6, 7, 9], [1/4, 1/4, 1/2], [6, 9], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], [6, 9], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
33. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[7, 10] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([7, 8, 10], [1/4, 1/4, 1/2], [7, 10], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], [7, 10], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
34. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[8, 11] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([8, 9, 11], [1/4, 1/4, 1/2], [8, 11], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], [8, 11], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
35. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[9, 12] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([9, 10, 12], [1/4, 1/4, 1/2], [9, 12], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], [9, 12], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
36. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[10, 13] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([10, 11, 13], [1/4, 1/4, 1/2], [10, 13], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/2], [10, 13], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
37. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[11, 14] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([11, 12, 14], [1/4, 1/4, 1/2], [11, 14], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], [11, 14], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
38. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[12, 15] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([12, 13, 15], [1/4, 1/4, 1/2], [12, 15], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], [12, 15], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
39. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[13, 16] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([13, 14, 16], [1/4, 1/4, 1/2], [13, 16], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], [13, 16], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
40. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[14, 17] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([14, 15, 17], [1/4, 1/4, 1/2], [14, 17], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], [14, 17], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
41. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[15, 18] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([15, 16, 18], [1/4, 1/4, 1/2], [15, 18], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/2], [15, 18], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
42. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[16, 19] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([16, 17, 19], [1/4, 1/4, 1/2], [16, 19], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], [16, 19], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
43. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[17, 20] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([17, 18, 20], [1/4, 1/4, 1/2], [17, 20], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], [17, 20], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
44. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[18, 21] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([18, 19, 21], [1/4, 1/4, 1/2], [18, 21], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], [18, 21], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
45. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[19, 22] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([19, 20, 22], [1/4, 1/4, 1/2], [19, 22], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], [19, 22], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
46. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[20, 23] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([20, 21, 23], [1/4, 1/4, 1/2], [20, 23], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/2], [20, 23], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
47. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[21, 24] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([21, 22, 24], [1/4, 1/4, 1/2], [21, 24], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], [21, 24], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
48. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[22, 25] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([22, 23, 25], [1/4, 1/4, 1/2], [22, 25], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], [22, 25], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
49. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[23, 26] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([23, 24, 26], [1/4, 1/4, 1/2], [23, 26], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], [23, 26], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
50. Sum Only the PMF Masses Whose Outcomes Are in the Event
Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Event input=[24, 27] is normalized to sorted distinct integers; retain listed values outside the outcome list in that event field but assign them no additional mass. Give (original outcomes,masses,normalized event,event probability).
View answer and explanation
([24, 25, 27], [1/4, 1/4, 1/2], [24, 27], 3/4)
Sum the aligned masses once for outcomes in the event; duplicates do not multiply probability and unlisted values contribute zero. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], [24, 27], 3/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-event-probability · candidate
51. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([0, 1, 3], [1/4, 1/4, 1/2], 7/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/2], 7/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
52. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([1, 2, 4], [1/4, 1/4, 1/2], 11/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], 11/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
53. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([2, 3, 5], [1/4, 1/4, 1/2], 15/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], 15/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
54. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([3, 4, 6], [1/4, 1/4, 1/2], 19/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], 19/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
55. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([4, 5, 7], [1/4, 1/4, 1/2], 23/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], 23/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
56. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([5, 6, 8], [1/4, 1/4, 1/2], 27/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/2], 27/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
57. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([6, 7, 9], [1/4, 1/4, 1/2], 31/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], 31/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
58. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([7, 8, 10], [1/4, 1/4, 1/2], 35/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], 35/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
59. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([8, 9, 11], [1/4, 1/4, 1/2], 39/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], 39/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
60. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([9, 10, 12], [1/4, 1/4, 1/2], 43/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], 43/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
61. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([10, 11, 13], [1/4, 1/4, 1/2], 47/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/2], 47/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
62. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([11, 12, 14], [1/4, 1/4, 1/2], 51/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], 51/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
63. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([12, 13, 15], [1/4, 1/4, 1/2], 55/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], 55/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
64. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([13, 14, 16], [1/4, 1/4, 1/2], 59/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], 59/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
65. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([14, 15, 17], [1/4, 1/4, 1/2], 63/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], 63/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
66. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([15, 16, 18], [1/4, 1/4, 1/2], 67/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/2], 67/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
67. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([16, 17, 19], [1/4, 1/4, 1/2], 71/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], 71/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
68. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([17, 18, 20], [1/4, 1/4, 1/2], 75/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], 75/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
69. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([18, 19, 21], [1/4, 1/4, 1/2], 79/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], 79/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
70. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([19, 20, 22], [1/4, 1/4, 1/2], 83/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], 83/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
71. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([20, 21, 23], [1/4, 1/4, 1/2], 87/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/2], 87/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
72. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([21, 22, 24], [1/4, 1/4, 1/2], 91/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], 91/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
73. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([22, 23, 25], [1/4, 1/4, 1/2], 95/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], 95/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
74. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([23, 24, 26], [1/4, 1/4, 1/2], 99/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], 99/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
75. An Exact Finite Weighted Mean, Not an Empirical Fit
Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Give (original outcomes,masses,sum of outcome*mass). Negative outcome values and a negative expectation are allowed.
View answer and explanation
([24, 25, 27], [1/4, 1/4, 1/2], 103/4)
Take the exact weighted sum using the original value-mass alignment. This is not an empirical fit, statistical estimate or validation of a population model. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], 103/4).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-expectation · candidate
76. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[0, 1, 3] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([0, 1, 3], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([0, 1, 3], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
77. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[1, 2, 4] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([1, 2, 4], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([1, 2, 4], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
78. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[2, 3, 5] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([2, 3, 5], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([2, 3, 5], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
79. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[3, 4, 6] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([3, 4, 6], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([3, 4, 6], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
80. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[4, 5, 7] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([4, 5, 7], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([4, 5, 7], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
81. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[5, 6, 8] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([5, 6, 8], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([5, 6, 8], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
82. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[6, 7, 9] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([6, 7, 9], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([6, 7, 9], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
83. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[7, 8, 10] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([7, 8, 10], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([7, 8, 10], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
84. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[8, 9, 11] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([8, 9, 11], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
85. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[9, 10, 12] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([9, 10, 12], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([9, 10, 12], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
86. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[10, 11, 13] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([10, 11, 13], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([10, 11, 13], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
87. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[11, 12, 14] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([11, 12, 14], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([11, 12, 14], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
88. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[12, 13, 15] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([12, 13, 15], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([12, 13, 15], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
89. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[13, 14, 16] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([13, 14, 16], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([13, 14, 16], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
90. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[14, 15, 17] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([14, 15, 17], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([14, 15, 17], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
91. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[15, 16, 18] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([15, 16, 18], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([15, 16, 18], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
92. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[16, 17, 19] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([16, 17, 19], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([16, 17, 19], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
93. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[17, 18, 20] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([17, 18, 20], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([17, 18, 20], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
94. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[18, 19, 21] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([18, 19, 21], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([18, 19, 21], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
95. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[19, 20, 22] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([19, 20, 22], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([19, 20, 22], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
96. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[20, 21, 23] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([20, 21, 23], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([20, 21, 23], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
97. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[21, 22, 24] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([21, 22, 24], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([21, 22, 24], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
98. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[22, 23, 25] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[1, 2, 1] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([22, 23, 25], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([22, 23, 25], [1/4, 1/4, 1/2], [1, 2, 1], [[1, 3/4], [2, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
99. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[23, 24, 26] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[2, 0, 2] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([23, 24, 26], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([23, 24, 26], [1/4, 1/4, 1/2], [2, 0, 2], [[0, 1/4], [2, 3/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
100. Merge All Masses with a Common Image
Use the explicit finite discrete outcome list=[24, 25, 27] and aligned exact masses=[1/4, 1/4, 1/2]. Outcomes are distinct; preserve this original order and its mass pairing. Use exact rationals, normalize denominator signs, reduce fractions and omit /1 for integer outputs. Qualification requires every mass nonnegative and total exactly 1. A mathematical distribution is not evidence of empirical fit, estimation quality or a real population. Require a qualified PMF. Images=[0, 1, 0] align with the original outcomes. Give (original outcomes,masses,original image list,every [image,combined mass] row in increasing image order). Retain even a listed image whose combined mass is zero.
View answer and explanation
([24, 25, 27], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]])
Group by image and add all corresponding rational masses. Preserve input alignment before sorting the distinct output images; this is a mathematical transformation, not estimation. This explicit exact slice is not a general theorem or empirical assertion. The exact result, retaining the requested order, is ([24, 25, 27], [1/4, 1/4, 1/2], [0, 1, 0], [[0, 3/4], [1, 1/4]]).
Reviewed step-by-step derivation
Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate
Canonical question JSONL · English JSONL · Validation report