# 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0001/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0002/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0003/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0004/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0005/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0006/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0007/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0008/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0009/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0010/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0011/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0012/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0013/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0014/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0015/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0016/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0017/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0018/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0019/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0020/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0021/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0022/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0023/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0024/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0025/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0026/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0027/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0028/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0029/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0030/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0031/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0032/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0033/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0034/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0035/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0036/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0037/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0038/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0039/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0040/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0041/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0042/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0043/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0044/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0045/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0046/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0047/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0048/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0049/) 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0050/) 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 explanation0/1 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0051/) 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 explanation4/1 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0052/) 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 explanation104/9 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0053/) 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 explanation908/49 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0054/) 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 explanation423/16 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0055/) 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 explanation2834/81 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0056/) 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 explanation44/1 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0057/) 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 explanation6456/121 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0058/) 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 explanation2267/36 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0059/) 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 explanation12302/169 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0060/) 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 explanation4056/49 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0061/) 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 explanation836/9 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0062/) 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 explanation6599/64 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0063/) 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 explanation32778/289 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0064/) 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 explanation10028/81 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0065/) 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 explanation48464/361 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0066/) 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 explanation579/4 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0067/) 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 explanation68486/441 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0068/) 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 explanation20072/121 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0069/) 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 explanation93372/529 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0070/) 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 explanation26951/144 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0071/) 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 explanation4946/25 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0072/) 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 explanation35244/169 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0073/) 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 explanation159848/729 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0074/) 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 explanation45083/196 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0075/) 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 explanationnone 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0076/) 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 explanation1 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0077/) 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 explanation2 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0078/) 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 explanation3 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0079/) 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 explanation4 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0080/) 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 explanation5 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0081/) 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 explanation6 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0082/) 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 explanation7 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0083/) 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 explanation2 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0084/) 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 explanation3 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0085/) 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 explanation4 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0086/) 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 explanation5 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0087/) 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 explanation6 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0088/) 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 explanation7 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0089/) 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 explanation3 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0090/) 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 explanation4 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0091/) 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 explanation5 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0092/) 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 explanation6 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0093/) 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 explanation7 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0094/) 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 explanation4 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0095/) 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 explanation5 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0096/) 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 explanation6 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0097/) 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 explanation7 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0098/) 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 explanation5 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0099/) 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 explanation6 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](https://ocme.evemisslab.com/en/questions/q-20261008-advance-20261010-r06-variance-0100/) Variance in finite probability models · qf-density10-variance-first-support-mismatch · candidate [Canonical question JSONL](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r06-variance-foundations/questions.jsonl) · [English JSONL](https://ocme.evemisslab.com/data/i18n/questions/batch-20261008-advance-20261010-r06-variance-foundations.en.jsonl) · [Validation report](https://ocme.evemisslab.com/data/questions/batches/batch-20261008-advance-20261010-r06-variance-foundations/validation-report.json)