OCMEBilingual mathematics & derivations繁體中文

Centered Probability Contributions and Positive Support

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-r06-variance-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 Centered Variance 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=[4, 99, 4] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[0/1, 0/1, 0/1]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [0/1, 0/1, 0/1].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

2. Every Centered Variance 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, 2] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[1/2, 1/2]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1/2, 1/2].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

3. Every Centered Variance 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, 3] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[169/72, 121/108, 289/216]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [169/72, 121/108, 289/216].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

4. Every Centered Variance 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, 4] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[1156/343, 648/343, 1024/343]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1156/343, 648/343, 1024/343].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

5. Every Centered Variance 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, 5] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[2205/512, 729/256, 2601/512]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [2205/512, 729/256, 2601/512].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

6. Every Centered Variance 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, 6] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[1250/243, 2888/729, 5476/729]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1250/243, 2888/729, 5476/729].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

7. Every Centered Variance 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, 7] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[5887/1000, 2601/500, 10201/1000]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [5887/1000, 2601/500, 10201/1000].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

8. Every Centered Variance 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, 8] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[72/11, 72/11, 144/11]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [72/11, 72/11, 144/11].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

9. Every Centered Variance 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, 9] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[1369/192, 6889/864, 27889/1728]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [1369/192, 6889/864, 27889/1728].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

10. Every Centered Variance 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, 10] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[16810/2197, 20808/2197, 42436/2197]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [16810/2197, 20808/2197, 42436/2197].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

11. Every Centered Variance 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, 11] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[22275/2744, 15129/1372, 62001/2744]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [22275/2744, 15129/1372, 62001/2744].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

12. Every Centered Variance 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, 12] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[9604/1125, 42632/3375, 87616/3375]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [9604/1125, 42632/3375, 87616/3375].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

13. Every Centered Variance 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, 13] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[36517/4096, 29241/2048, 120409/4096]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [36517/4096, 29241/2048, 120409/4096].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

14. Every Centered Variance 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, 14] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[45486/4913, 78408/4913, 161604/4913]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [45486/4913, 78408/4913, 161604/4913].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

15. Every Centered Variance 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, 15] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[18605/1944, 51529/2916, 212521/5832]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [18605/1944, 51529/2916, 212521/5832].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

16. Every Centered Variance 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, 16] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[67600/6859, 133128/6859, 274576/6859]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [67600/6859, 133128/6859, 274576/6859].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

17. Every Centered Variance 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, 17] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[80937/8000, 84681/4000, 349281/8000]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [80937/8000, 84681/4000, 349281/8000].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

18. Every Centered Variance 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, 18] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[10658/1029, 212552/9261, 438244/9261]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [10658/1029, 212552/9261, 438244/9261].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

19. Every Centered Variance 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, 19] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[931/88, 99/4, 4489/88]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [931/88, 99/4, 4489/88].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

20. Every Centered Variance 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, 20] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[131220/12167, 323208/12167, 665856/12167]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [131220/12167, 323208/12167, 665856/12167].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

21. Every Centered Variance 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, 21] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[50575/4608, 196249/6912, 808201/13824]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [50575/4608, 196249/6912, 808201/13824].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

22. Every Centered Variance 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, 22] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[174262/15625, 472392/15625, 972196/15625]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [174262/15625, 472392/15625, 972196/15625].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

23. Every Centered Variance 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, 23] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[198927/17576, 281961/8788, 1159929/17576]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [198927/17576, 281961/8788, 1159929/17576].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

24. Every Centered Variance 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, 24] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[75272/6561, 668168/19683, 1373584/19683]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [75272/6561, 668168/19683, 1373584/19683].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

25. Every Centered Variance 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, 25] in index order. First take μ=E[X]; list w_i(X_i−μ)² in index order, without omitting zeros. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

[255025/21952, 393129/10976, 1615441/21952]

Find the exact mean, square each deviation, then weight it; a differing zero-weight value still contributes zero. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is [255025/21952, 393129/10976, 1615441/21952].

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-contributions · candidate

26. Second Moment and Squared 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=[4, 99, 4] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(16/1, 16/1)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (16/1, 16/1).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

27. Second Moment and Squared Mean

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=[-1, 1] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(1/1, 0/1)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (1/1, 0/1).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

28. Second Moment and Squared 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, 2, 3] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(29/6, 1/36)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (29/6, 1/36).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

29. Second Moment and Squared 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=[-3, 2, 4] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(60/7, 16/49)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (60/7, 16/49).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

30. Second Moment and Squared 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, 2, 5] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(113/8, 121/64)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (113/8, 121/64).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

31. Second Moment and Squared 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, 2, 6] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(194/9, 400/81)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (194/9, 400/81).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

32. Second Moment and Squared 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=[-6, 2, 7] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(309/10, 961/100)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (309/10, 961/100).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

33. Second Moment and Squared 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, 2, 8] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(464/11, 16/1)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (464/11, 16/1).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

34. Second Moment and Squared 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, 2, 9] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(665/12, 3481/144)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (665/12, 3481/144).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

35. Second Moment and Squared 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=[-9, 2, 10] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(918/13, 5776/169)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (918/13, 5776/169).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

36. Second Moment and Squared 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, 2, 11] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(1229/14, 9025/196)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (1229/14, 9025/196).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

37. Second Moment and Squared 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, 2, 12] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(1604/15, 13456/225)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (1604/15, 13456/225).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

38. Second Moment and Squared 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=[-12, 2, 13] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(2049/16, 19321/256)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (2049/16, 19321/256).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

39. Second Moment and Squared 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, 2, 14] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(2570/17, 26896/289)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (2570/17, 26896/289).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

40. Second Moment and Squared 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, 2, 15] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(3173/18, 36481/324)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (3173/18, 36481/324).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

41. Second Moment and Squared 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=[-15, 2, 16] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(3864/19, 48400/361)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (3864/19, 48400/361).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

42. Second Moment and Squared 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, 2, 17] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(4649/20, 63001/400)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (4649/20, 63001/400).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

43. Second Moment and Squared 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, 2, 18] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(5534/21, 80656/441)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (5534/21, 80656/441).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

44. Second Moment and Squared 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=[-18, 2, 19] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(6525/22, 841/4)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (6525/22, 841/4).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

45. Second Moment and Squared 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, 2, 20] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(7628/23, 126736/529)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (7628/23, 126736/529).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

46. Second Moment and Squared 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, 2, 21] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(8849/24, 156025/576)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (8849/24, 156025/576).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

47. Second Moment and Squared 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=[-21, 2, 22] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(10194/25, 190096/625)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (10194/25, 190096/625).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

48. Second Moment and Squared 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, 2, 23] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(11669/26, 229441/676)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (11669/26, 229441/676).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

49. Second Moment and Squared 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, 2, 24] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(13280/27, 274576/729)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (13280/27, 274576/729).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

50. Second Moment and Squared 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=[-24, 2, 25] in index order. Give (E[X²], (E[X])²), distinguishing the two expressions. Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

(15033/28, 326041/784)

Square each value before weighting for the first field, and square the weighted mean for the second; their difference is the variance. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is (15033/28, 326041/784).

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-second-moment-pair · candidate

51. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

0/1

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 0/1.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

52. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

4/1

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4/1.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

53. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

104/9

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 104/9.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

54. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

908/49

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 908/49.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

55. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

423/16

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 423/16.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

56. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

2834/81

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2834/81.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

57. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

44/1

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 44/1.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

58. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

6456/121

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6456/121.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

59. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

2267/36

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2267/36.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

60. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

12302/169

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 12302/169.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

61. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

4056/49

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4056/49.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

62. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

836/9

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 836/9.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

63. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

6599/64

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6599/64.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

64. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

32778/289

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 32778/289.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

65. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

10028/81

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 10028/81.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

66. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

48464/361

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 48464/361.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

67. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

579/4

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 579/4.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

68. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

68486/441

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 68486/441.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

69. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

20072/121

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 20072/121.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

70. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

93372/529

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 93372/529.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

71. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

26951/144

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 26951/144.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

72. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

4946/25

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4946/25.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

73. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

35244/169

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 35244/169.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

74. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

159848/729

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 159848/729.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

75. Variance of a Shared-Model Difference

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 Var(X−Y), not Var(X)+Var(Y). Use exact reduced fractions with positive denominators, retaining /1 for integers.

View answer and explanation

45083/196

Subtract pointwise before computing the weighted variance; dependence may make the difference constant on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 45083/196.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-difference · candidate

76. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 2, 0, 1, 0, 0, 0, 0], D=3, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[99, 4, -99, 4, 9, 8, 7, 6] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

none

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is none.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

77. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[2, 2, 0, 0, 0, 0, 0, 0], D=4, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 2, 102, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

1

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 1.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

78. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 3, 2, 0, 0, 0, 0, 0], D=5, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 3, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

2

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

79. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 4, 2, 0, 0, 0, 0], D=6, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 4, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

3

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 3.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

80. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 5, 2, 0, 0, 0], D=7, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 5, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

4

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

81. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 6, 2, 0, 0], D=8, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 0, 6, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

5

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

82. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 0, 7, 2, 0], D=9, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 104, 0, 7, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

6

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

83. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 0, 0, 8, 2], D=10, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 104, 105, 0, 8] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

7

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

84. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[9, 0, 2, 0, 0, 0, 0, 0], D=11, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 9, 103, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

2

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 2.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

85. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 10, 0, 2, 0, 0, 0, 0], D=12, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 10, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

3

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 3.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

86. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 11, 0, 2, 0, 0, 0], D=13, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 103, 11, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

4

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

87. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 12, 0, 2, 0, 0], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 12, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

5

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

88. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 13, 0, 2, 0], D=15, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 0, 105, 13, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

6

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

89. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 0, 14, 0, 2], D=16, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 104, 0, 106, 14] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

7

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

90. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[15, 0, 0, 2, 0, 0, 0, 0], D=17, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 102, 15, 104, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

3

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 3.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

91. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 16, 0, 0, 2, 0, 0, 0], D=18, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 103, 16, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

4

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

92. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 17, 0, 0, 2, 0, 0], D=19, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 103, 104, 17, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

5

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

93. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 18, 0, 0, 2, 0], D=20, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 105, 18, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

6

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

94. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 0, 19, 0, 0, 2], D=21, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 103, 0, 105, 106, 19] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

7

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

95. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[20, 0, 0, 0, 2, 0, 0, 0], D=22, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 102, 103, 20, 105, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

4

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 4.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

96. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 21, 0, 0, 0, 2, 0, 0], D=23, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 103, 104, 21, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

5

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

97. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 22, 0, 0, 0, 2, 0], D=24, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 0, 103, 104, 105, 22, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

6

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

98. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 0, 0, 23, 0, 0, 0, 2], D=25, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 105, 106, 23] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

7

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 7.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

99. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[24, 0, 0, 0, 0, 2, 0, 0], D=26, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[0, 101, 102, 103, 104, 24, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

5

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 5.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

100. First Mismatch on Positive Probability Support

S={0,…,7}, F=P(S), masses=[0, 25, 0, 0, 0, 0, 2, 0], D=27, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 0, 102, 103, 104, 105, 25, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0.

View answer and explanation

6

Ignore differing zero-weight values when checking constancy on positive support. These are finite probability quantities, not sample-variance estimates. The exact result, retaining the requested order, is 6.

Reviewed step-by-step derivation

Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate

Canonical question JSONL · English JSONL · Validation report