# 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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261013-advance-20261022-r07-finite-discrete-probability-distribution-0100/)

Finite discrete probability distribution · qf-str22-distribution-pushforward · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261013-advance-20261022-r07-finite-discrete-probability-distribution-foundations/validation-report.json)