# Shared-Model Expectations and Positive-Mass Recovery

This batch has passed Schema, deduplication, source-snapshot, and independent-answer recomputation checks; it is still not marked as human-reviewed.

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-20261008-advance-20261010-r05-expectation-foundations · 2026-10-08 · 100 questions

This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.



## 1. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 99, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-1/1, 0/1, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-1/1, 0/1, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0001/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 2. Every Weighted Expectation Contribution

S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[0/1, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [0/1, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0002/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 3. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[3, 2, 1], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-1/1, 2/3, 1/6]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-1/1, 2/3, 1/6].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0003/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 4. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-12/7, 4/7, -1/7]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-12/7, 4/7, -1/7].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0004/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 5. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-5/2, 1/2, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-5/2, 1/2, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0005/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 6. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-10/3, 4/9, 1/9]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-10/3, 4/9, 1/9].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0006/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 7. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-6, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-21/5, 2/5, -1/10]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-21/5, 2/5, -1/10].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0007/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 8. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-56/11, 4/11, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-56/11, 4/11, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0008/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 9. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-6/1, 1/3, 1/12]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-6/1, 1/3, 1/12].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0009/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 10. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-9, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-90/13, 4/13, -1/13]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-90/13, 4/13, -1/13].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0010/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 11. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-55/7, 2/7, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-55/7, 2/7, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0011/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 12. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-44/5, 4/15, 1/15]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-44/5, 4/15, 1/15].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0012/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 13. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-39/4, 1/4, -1/16]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-39/4, 1/4, -1/16].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0013/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 14. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-182/17, 4/17, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-182/17, 4/17, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0014/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 15. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-35/3, 2/9, 1/18]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-35/3, 2/9, 1/18].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0015/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 16. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-15, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-240/19, 4/19, -1/19]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-240/19, 4/19, -1/19].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0016/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 17. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-68/5, 1/5, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-68/5, 1/5, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0017/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 18. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-102/7, 4/21, 1/21]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-102/7, 4/21, 1/21].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0018/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 19. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-18, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-171/11, 2/11, -1/22]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-171/11, 2/11, -1/22].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0019/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 20. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-380/23, 4/23, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-380/23, 4/23, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0020/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 21. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-35/2, 1/6, 1/24]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-35/2, 1/6, 1/24].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0021/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 22. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-462/25, 4/25, -1/25]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-462/25, 4/25, -1/25].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0022/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 23. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 0] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-253/13, 2/13, 0/1]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-253/13, 2/13, 0/1].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0023/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 24. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-184/9, 4/27, 1/27]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-184/9, 4/27, 1/27].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0024/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 25. Every Weighted Expectation Contribution

S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, -1] in index order. List w_i X_i in index order from 0, retaining zero terms. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation[-150/7, 1/7, -1/28]

Multiply each mass by its value and divide by D; zero-weight values contribute zero, and negative values can contribute negatively. The exact result, retaining the requested order, is [-150/7, 1/7, -1/28].



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0025/)

Expectation in finite models · qf-density10-expectation-contributions · candidate





## 26. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[4, 99, 4] in index order and Y=[4, -50, 4] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation0/1

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0026/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 27. Expectation Difference on a Shared Model

S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 2] in index order and Y=[2, 0] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation0/1

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0027/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 28. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[3, 2, 1], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 2, 3] in index order and Y=[1, -1, -2] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation1/3

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is 1/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0028/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 29. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 2, 4] in index order and Y=[1, -1, -3] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-3/7

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -3/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0029/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 30. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 5] in index order and Y=[1, -1, -4] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-5/4

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -5/4.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0030/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 31. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 6] in index order and Y=[1, -1, -5] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-19/9

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -19/9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0031/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 32. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-6, 2, 7] in index order and Y=[1, -1, -6] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-3/1

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -3/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0032/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 33. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 8] in index order and Y=[1, -1, -7] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-43/11

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -43/11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0033/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 34. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 9] in index order and Y=[1, -1, -8] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-29/6

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -29/6.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0034/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 35. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-9, 2, 10] in index order and Y=[1, -1, -9] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-75/13

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -75/13.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0035/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 36. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 11] in index order and Y=[1, -1, -10] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-47/7

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -47/7.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0036/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 37. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 12] in index order and Y=[1, -1, -11] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-23/3

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -23/3.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0037/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 38. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[1, -1, -12] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-69/8

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -69/8.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0038/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 39. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, 2, 14] in index order and Y=[1, -1, -13] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-163/17

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -163/17.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0039/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 40. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 15] in index order and Y=[1, -1, -14] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-95/9

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -95/9.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0040/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 41. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-15, 2, 16] in index order and Y=[1, -1, -15] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-219/19

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -219/19.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0041/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 42. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, 2, 17] in index order and Y=[1, -1, -16] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-25/2

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -25/2.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0042/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 43. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 2, 18] in index order and Y=[1, -1, -17] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-283/21

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -283/21.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0043/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 44. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-18, 2, 19] in index order and Y=[1, -1, -18] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-159/11

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -159/11.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0044/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 45. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 20] in index order and Y=[1, -1, -19] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-355/23

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -355/23.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0045/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 46. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[1, -1, -20] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-197/12

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -197/12.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0046/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 47. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, 22] in index order and Y=[1, -1, -21] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-87/5

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -87/5.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0047/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 48. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 23] in index order and Y=[1, -1, -22] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-239/13

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -239/13.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0048/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 49. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] in index order and Y=[1, -1, -23] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-523/27

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -523/27.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0049/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 50. Expectation Difference on a Shared Model

S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, 25] in index order and Y=[1, -1, -24] on the same S and weights, with no independence premise. Find E[X]−E[Y], equivalently E[X−Y]. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation-285/14

Compute the two weighted means and subtract; linearity on the shared model does not require independence. The exact result, retaining the requested order, is -285/14.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0050/)

Expectation in finite models · qf-density10-expectation-difference · candidate





## 51. Product Expectation versus Product of Means

S={0,…,1}, F=P(S), masses=[1, 1], D=2, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 1] in index order and Y=[0, 1] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(1/2, 1/4)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (1/2, 1/4).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0051/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 52. Product Expectation versus Product of Means

S={0,…,1}, F=P(S), masses=[1, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[3, 3] in index order and Y=[-1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(3/1, 3/1)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (3/1, 3/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0052/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 53. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 99, 1] in index order and Y=[1, -100, -2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2/1, 0/1)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-2/1, 0/1).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0053/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 54. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-3, 2, 4] in index order and Y=[5, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-8/1, -80/49)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-8/1, -80/49).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0054/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 55. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, 2, 5] in index order and Y=[6, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-57/4, -165/32)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-57/4, -165/32).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0055/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 56. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 2, 6] in index order and Y=[7, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-202/9, -280/27)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-202/9, -280/27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0056/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 57. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-6, 2, 7] in index order and Y=[8, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-163/5, -434/25)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-163/5, -434/25).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0057/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 58. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, 2, 8] in index order and Y=[9, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-492/11, -288/11)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-492/11, -288/11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0058/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 59. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 2, 9] in index order and Y=[10, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-353/6, -295/8)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-353/6, -295/8).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0059/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 60. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-9, 2, 10] in index order and Y=[11, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-974/13, -8360/169)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-974/13, -8360/169).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0060/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 61. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, 2, 11] in index order and Y=[12, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-93/1, -3135/49)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-93/1, -3135/49).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0061/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 62. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 2, 12] in index order and Y=[13, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1696/15, -6032/75)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1696/15, -6032/75).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0062/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 63. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-12, 2, 13] in index order and Y=[14, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1081/8, -12649/128)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1081/8, -12649/128).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0063/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 64. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, 2, 14] in index order and Y=[15, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2706/17, -34440/289)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-2706/17, -34440/289).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0064/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 65. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 2, 15] in index order and Y=[16, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1667/9, -3820/27)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1667/9, -3820/27).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0065/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 66. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-15, 2, 16] in index order and Y=[17, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4052/19, -59840/361)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-4052/19, -59840/361).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0066/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 67. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, 2, 17] in index order and Y=[18, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-2433/10, -38403/200)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-2433/10, -38403/200).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0067/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 68. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 2, 18] in index order and Y=[19, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-826/3, -10792/49)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-826/3, -10792/49).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0068/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 69. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-18, 2, 19] in index order and Y=[20, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-3403/11, -2755/11)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-3403/11, -2755/11).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0069/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 70. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, 2, 20] in index order and Y=[21, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-7944/23, -149520/529)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-7944/23, -149520/529).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0070/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 71. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 2, 21] in index order and Y=[22, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-4601/12, -30415/96)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-4601/12, -30415/96).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0071/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 72. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-21, 2, 22] in index order and Y=[23, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-10586/25, -220616/625)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-10586/25, -220616/625).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0072/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 73. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, 2, 23] in index order and Y=[24, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-6051/13, -66102/169)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-6051/13, -66102/169).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0073/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 74. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 2, 24] in index order and Y=[25, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-13756/27, -104800/243)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-13756/27, -104800/243).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0074/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 75. Product Expectation versus Product of Means

S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-24, 2, 25] in index order and Y=[26, -1, 2] on the same S and weights, with no independence premise. Give (E[XY], E[X]E[Y]). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation(-1111/2, -185575/392)

Weight pointwise products for the first field, and multiply the two means for the second; numerical equality alone does not imply independence. The exact result, retaining the requested order, is (-1111/2, -185575/392).



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0075/)

Expectation in finite models · qf-density10-expectation-product-pair · candidate





## 76. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[1, 0, 2], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 99, 3] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=1/1. Find that slot’s integer value, written as p/1.

View answer and explanation-3/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -3/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0076/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 77. Positive-Mass Value Recovered from a Mean

S={0,…,0}, F=P(S), masses=[1], D=1, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=0/1. Find that slot’s integer value, written as p/1.

View answer and explanation0/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0077/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 78. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[3, 2, 1], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-2, 4, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=1/3. Find that slot’s integer value, written as p/1.

View answer and explanation0/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 0/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0078/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 79. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[4, 2, 1], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 5, -1] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-3/7. Find that slot’s integer value, written as p/1.

View answer and explanation-3/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -3/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0079/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 80. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[5, 2, 1], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-4, null, -2] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-5/4. Find that slot’s integer value, written as p/1.

View answer and explanation6/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0080/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 81. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[6, 2, 1], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-5, 7, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-19/9. Find that slot’s integer value, written as p/1.

View answer and explanation-3/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -3/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0081/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 82. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[7, 2, 1], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 8, -4] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-3/1. Find that slot’s integer value, written as p/1.

View answer and explanation-6/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0082/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 83. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[8, 2, 1], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-7, null, -5] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-43/11. Find that slot’s integer value, written as p/1.

View answer and explanation9/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 9/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0083/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 84. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[9, 2, 1], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-8, 10, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-29/6. Find that slot’s integer value, written as p/1.

View answer and explanation-6/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -6/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0084/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 85. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[10, 2, 1], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 11, -7] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-75/13. Find that slot’s integer value, written as p/1.

View answer and explanation-9/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -9/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0085/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 86. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[11, 2, 1], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-10, null, -8] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-47/7. Find that slot’s integer value, written as p/1.

View answer and explanation12/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 12/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0086/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 87. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[12, 2, 1], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-11, 13, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-23/3. Find that slot’s integer value, written as p/1.

View answer and explanation-9/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -9/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0087/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 88. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[13, 2, 1], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 14, -10] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-69/8. Find that slot’s integer value, written as p/1.

View answer and explanation-12/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -12/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0088/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 89. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[14, 2, 1], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-13, null, -11] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-163/17. Find that slot’s integer value, written as p/1.

View answer and explanation15/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 15/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0089/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 90. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[15, 2, 1], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-14, 16, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-95/9. Find that slot’s integer value, written as p/1.

View answer and explanation-12/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -12/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0090/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 91. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[16, 2, 1], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 17, -13] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-219/19. Find that slot’s integer value, written as p/1.

View answer and explanation-15/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -15/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0091/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 92. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[17, 2, 1], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-16, null, -14] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-25/2. Find that slot’s integer value, written as p/1.

View answer and explanation18/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 18/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0092/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 93. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[18, 2, 1], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-17, 19, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-283/21. Find that slot’s integer value, written as p/1.

View answer and explanation-15/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -15/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0093/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 94. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[19, 2, 1], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 20, -16] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-159/11. Find that slot’s integer value, written as p/1.

View answer and explanation-18/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -18/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0094/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 95. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[20, 2, 1], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-19, null, -17] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-355/23. Find that slot’s integer value, written as p/1.

View answer and explanation21/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 21/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0095/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 96. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[21, 2, 1], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-20, 22, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-197/12. Find that slot’s integer value, written as p/1.

View answer and explanation-18/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -18/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0096/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 97. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[22, 2, 1], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 23, -19] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-87/5. Find that slot’s integer value, written as p/1.

View answer and explanation-21/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -21/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0097/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 98. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[23, 2, 1], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-22, null, -20] in index order. null marks the unique unknown slot, k=1, whose mass is positive. E[X]=-239/13. Find that slot’s integer value, written as p/1.

View answer and explanation24/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is 24/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0098/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 99. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[24, 2, 1], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[-23, 25, null] in index order. null marks the unique unknown slot, k=2, whose mass is positive. E[X]=-523/27. Find that slot’s integer value, written as p/1.

View answer and explanation-21/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -21/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0099/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate





## 100. Positive-Mass Value Recovered from a Mean

S={0,…,2}, F=P(S), masses=[25, 2, 1], D=28, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[null, 26, -22] in index order. null marks the unique unknown slot, k=0, whose mass is positive. E[X]=-285/14. Find that slot’s integer value, written as p/1.

View answer and explanation-24/1

Multiply the target mean by D, subtract the known weighted sum, then divide by the unknown slot mass. Positive mass permits recovery, and this instance has an integer solution. The exact result, retaining the requested order, is -24/1.



[Reviewed step-by-step derivation](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r05-expectation-0100/)

Expectation in finite models · qf-density10-expectation-missing-value · candidate



[Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r05-expectation-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261010-r05-expectation-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r05-expectation-foundations/validation-report.json)