Shared-Model Expectations and Positive-Mass Recovery
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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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
0/1
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
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 explanation
0/1
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
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 explanation
1/3
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 explanation
0/1
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
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 explanation
0/1
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
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
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 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
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
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
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 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
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
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
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 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
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
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
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 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
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
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
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 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
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
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
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 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
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
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
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 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
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
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
Expectation in finite models · qf-density10-expectation-missing-value · candidate
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