Complete Partitions, Exact Cell Means, Null Cells and the Tower Identity
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batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
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1. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/10], [1, 7/10]], Yes, 23/10)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/10], [1, 7/10]], Yes, 23/10).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
2. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/11], [1, 8/11]], Yes, 37/11)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/11], [1, 8/11]], Yes, 37/11).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
3. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/4], [1, 3/4]], Yes, 53/12)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/4], [1, 3/4]], Yes, 53/12).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
4. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/13], [1, 10/13]], Yes, 71/13)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/13], [1, 10/13]], Yes, 71/13).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
5. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/14], [1, 11/14]], Yes, 13/2)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/14], [1, 11/14]], Yes, 13/2).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
6. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/5], [1, 4/5]], Yes, 113/15)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/5], [1, 4/5]], Yes, 113/15).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
7. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/16], [1, 13/16]], Yes, 137/16)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/16], [1, 13/16]], Yes, 137/16).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
8. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/17], [1, 14/17]], Yes, 163/17)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/17], [1, 14/17]], Yes, 163/17).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
9. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/6], [1, 5/6]], Yes, 191/18)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/6], [1, 5/6]], Yes, 191/18).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
10. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/19], [1, 16/19]], Yes, 221/19)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/19], [1, 16/19]], Yes, 221/19).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
11. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/20], [1, 17/20]], Yes, 253/20)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/20], [1, 17/20]], Yes, 253/20).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
12. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/7], [1, 6/7]], Yes, 41/3)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/7], [1, 6/7]], Yes, 41/3).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
13. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/22], [1, 19/22]], Yes, 323/22)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/22], [1, 19/22]], Yes, 323/22).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
14. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/23], [1, 20/23]], Yes, 361/23)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/23], [1, 20/23]], Yes, 361/23).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
15. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/8], [1, 7/8]], Yes, 401/24)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/8], [1, 7/8]], Yes, 401/24).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
16. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/25], [1, 22/25]], Yes, 443/25)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/25], [1, 22/25]], Yes, 443/25).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
17. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/26], [1, 23/26]], Yes, 487/26)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/26], [1, 23/26]], Yes, 487/26).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
18. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/9], [1, 8/9]], Yes, 533/27)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/9], [1, 8/9]], Yes, 533/27).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
19. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/28], [1, 25/28]], Yes, 83/4)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/28], [1, 25/28]], Yes, 83/4).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
20. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/29], [1, 26/29]], Yes, 631/29)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/29], [1, 26/29]], Yes, 631/29).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
21. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/10], [1, 9/10]], Yes, 683/30)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/10], [1, 9/10]], Yes, 683/30).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
22. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/31], [1, 28/31]], Yes, 737/31)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/31], [1, 28/31]], Yes, 737/31).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
23. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/32], [1, 29/32]], Yes, 793/32)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/32], [1, 29/32]], Yes, 793/32).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
24. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 1/11], [1, 10/11]], Yes, 851/33)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 1/11], [1, 10/11]], Yes, 851/33).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
25. A Complete Disjoint Partition and Global Mean Use the Same Law
Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (partition, cell_masses, complete_disjoint_certificate, mean).
View answer and explanation
([[0, 1], [2, 3]], [[0, 3/34], [1, 31/34]], Yes, 911/34)
Check complete disjoint coverage, compute each cell mass in supplied order and the global exact expectation. Zero-mass atoms remain part of the partition even though they contribute zero to integrals. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, 3/34], [1, 31/34]], Yes, 911/34).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-partition-audit · candidate
26. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/10, 2/3, 1/5], [1, 7/10, 3, 21/10]], 23/10, 23/10, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/10, 2/3, 1/5], [1, 7/10, 3, 21/10]], 23/10, 23/10, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
27. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/11, 5/3, 5/11], [1, 8/11, 4, 32/11]], 37/11, 37/11, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/11, 5/3, 5/11], [1, 8/11, 4, 32/11]], 37/11, 37/11, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
28. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/4, 8/3, 2/3], [1, 3/4, 5, 15/4]], 53/12, 53/12, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/4, 8/3, 2/3], [1, 3/4, 5, 15/4]], 53/12, 53/12, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
29. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/13, 11/3, 11/13], [1, 10/13, 6, 60/13]], 71/13, 71/13, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/13, 11/3, 11/13], [1, 10/13, 6, 60/13]], 71/13, 71/13, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
30. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/14, 14/3, 1], [1, 11/14, 7, 11/2]], 13/2, 13/2, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/14, 14/3, 1], [1, 11/14, 7, 11/2]], 13/2, 13/2, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
31. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/5, 17/3, 17/15], [1, 4/5, 8, 32/5]], 113/15, 113/15, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/5, 17/3, 17/15], [1, 4/5, 8, 32/5]], 113/15, 113/15, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
32. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/16, 20/3, 5/4], [1, 13/16, 9, 117/16]], 137/16, 137/16, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/16, 20/3, 5/4], [1, 13/16, 9, 117/16]], 137/16, 137/16, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
33. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/17, 23/3, 23/17], [1, 14/17, 10, 140/17]], 163/17, 163/17, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
34. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/6, 26/3, 13/9], [1, 5/6, 11, 55/6]], 191/18, 191/18, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/6, 26/3, 13/9], [1, 5/6, 11, 55/6]], 191/18, 191/18, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
35. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/19, 29/3, 29/19], [1, 16/19, 12, 192/19]], 221/19, 221/19, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/19, 29/3, 29/19], [1, 16/19, 12, 192/19]], 221/19, 221/19, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
36. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/20, 32/3, 8/5], [1, 17/20, 13, 221/20]], 253/20, 253/20, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/20, 32/3, 8/5], [1, 17/20, 13, 221/20]], 253/20, 253/20, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
37. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/7, 35/3, 5/3], [1, 6/7, 14, 12]], 41/3, 41/3, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/7, 35/3, 5/3], [1, 6/7, 14, 12]], 41/3, 41/3, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
38. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/22, 38/3, 19/11], [1, 19/22, 15, 285/22]], 323/22, 323/22, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/22, 38/3, 19/11], [1, 19/22, 15, 285/22]], 323/22, 323/22, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
39. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/23, 41/3, 41/23], [1, 20/23, 16, 320/23]], 361/23, 361/23, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/23, 41/3, 41/23], [1, 20/23, 16, 320/23]], 361/23, 361/23, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
40. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/8, 44/3, 11/6], [1, 7/8, 17, 119/8]], 401/24, 401/24, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
41. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/25, 47/3, 47/25], [1, 22/25, 18, 396/25]], 443/25, 443/25, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/25, 47/3, 47/25], [1, 22/25, 18, 396/25]], 443/25, 443/25, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
42. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/26, 50/3, 25/13], [1, 23/26, 19, 437/26]], 487/26, 487/26, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
43. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/9, 53/3, 53/27], [1, 8/9, 20, 160/9]], 533/27, 533/27, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/9, 53/3, 53/27], [1, 8/9, 20, 160/9]], 533/27, 533/27, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
44. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/28, 56/3, 2], [1, 25/28, 21, 75/4]], 83/4, 83/4, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/28, 56/3, 2], [1, 25/28, 21, 75/4]], 83/4, 83/4, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
45. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/29, 59/3, 59/29], [1, 26/29, 22, 572/29]], 631/29, 631/29, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/29, 59/3, 59/29], [1, 26/29, 22, 572/29]], 631/29, 631/29, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
46. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/10, 62/3, 31/15], [1, 9/10, 23, 207/10]], 683/30, 683/30, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/10, 62/3, 31/15], [1, 9/10, 23, 207/10]], 683/30, 683/30, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
47. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/31, 65/3, 65/31], [1, 28/31, 24, 672/31]], 737/31, 737/31, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/31, 65/3, 65/31], [1, 28/31, 24, 672/31]], 737/31, 737/31, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
48. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/32, 68/3, 17/8], [1, 29/32, 25, 725/32]], 793/32, 793/32, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/32, 68/3, 17/8], [1, 29/32, 25, 725/32]], 793/32, 793/32, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
49. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 1/11, 71/3, 71/33], [1, 10/11, 26, 260/11]], 851/33, 851/33, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 1/11, 71/3, 71/33], [1, 10/11, 26, 260/11]], 851/33, 851/33, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
50. Weight Every Positive Cell Mean by Its Own Cell Probability
Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, weighted_sum, mean, equal).
View answer and explanation
([[0, 3/34, 74/3, 37/17], [1, 31/34, 27, 837/34]], 911/34, 911/34, Yes)
Require every cell to have positive mass. Each cell row is [cell index,cell mass,conditional mean,weighted contribution]. Sum those contributions and compare with the global expectation; do not take an unweighted average of cell means. The exact result, retaining the requested order, is ([[0, 3/34, 74/3, 37/17], [1, 31/34, 27, 837/34]], 911/34, 911/34, Yes).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-positive-cells-formula · candidate
51. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([2/3, 2/3, 3, 3], 23/10, 23/10, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([2/3, 2/3, 3, 3], 23/10, 23/10, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
52. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([5/3, 5/3, 4, 4], 37/11, 37/11, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([5/3, 5/3, 4, 4], 37/11, 37/11, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
53. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([8/3, 8/3, 5, 5], 53/12, 53/12, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([8/3, 8/3, 5, 5], 53/12, 53/12, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
54. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([11/3, 11/3, 6, 6], 71/13, 71/13, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([11/3, 11/3, 6, 6], 71/13, 71/13, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
55. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([14/3, 14/3, 7, 7], 13/2, 13/2, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([14/3, 14/3, 7, 7], 13/2, 13/2, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
56. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([17/3, 17/3, 8, 8], 113/15, 113/15, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([17/3, 17/3, 8, 8], 113/15, 113/15, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
57. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([20/3, 20/3, 9, 9], 137/16, 137/16, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([20/3, 20/3, 9, 9], 137/16, 137/16, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
58. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([23/3, 23/3, 10, 10], 163/17, 163/17, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([23/3, 23/3, 10, 10], 163/17, 163/17, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
59. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([26/3, 26/3, 11, 11], 191/18, 191/18, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([26/3, 26/3, 11, 11], 191/18, 191/18, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
60. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([29/3, 29/3, 12, 12], 221/19, 221/19, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([29/3, 29/3, 12, 12], 221/19, 221/19, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
61. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([32/3, 32/3, 13, 13], 253/20, 253/20, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([32/3, 32/3, 13, 13], 253/20, 253/20, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
62. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([35/3, 35/3, 14, 14], 41/3, 41/3, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([35/3, 35/3, 14, 14], 41/3, 41/3, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
63. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([38/3, 38/3, 15, 15], 323/22, 323/22, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([38/3, 38/3, 15, 15], 323/22, 323/22, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
64. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([41/3, 41/3, 16, 16], 361/23, 361/23, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([41/3, 41/3, 16, 16], 361/23, 361/23, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
65. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([44/3, 44/3, 17, 17], 401/24, 401/24, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([44/3, 44/3, 17, 17], 401/24, 401/24, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
66. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([47/3, 47/3, 18, 18], 443/25, 443/25, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([47/3, 47/3, 18, 18], 443/25, 443/25, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
67. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([50/3, 50/3, 19, 19], 487/26, 487/26, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([50/3, 50/3, 19, 19], 487/26, 487/26, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
68. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([53/3, 53/3, 20, 20], 533/27, 533/27, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([53/3, 53/3, 20, 20], 533/27, 533/27, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
69. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([56/3, 56/3, 21, 21], 83/4, 83/4, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([56/3, 56/3, 21, 21], 83/4, 83/4, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
70. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([59/3, 59/3, 22, 22], 631/29, 631/29, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([59/3, 59/3, 22, 22], 631/29, 631/29, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
71. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([62/3, 62/3, 23, 23], 683/30, 683/30, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([62/3, 62/3, 23, 23], 683/30, 683/30, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
72. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([65/3, 65/3, 24, 24], 737/31, 737/31, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([65/3, 65/3, 24, 24], 737/31, 737/31, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
73. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([68/3, 68/3, 25, 25], 793/32, 793/32, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([68/3, 68/3, 25, 25], 793/32, 793/32, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
74. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([71/3, 71/3, 26, 26], 851/33, 851/33, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([71/3, 71/3, 26, 26], 851/33, 851/33, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
75. A Cell-Constant Conditional Version Has the Same Expectation
Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate).
On a null-mass cell this conditional-expectation version chooses0. It is not a uniquely determined conditional quotient and does not affect the weighted expectation.
View answer and explanation
([74/3, 74/3, 27, 27], 911/34, 911/34, Yes, conditional_expectation_constant_on_partition_cells)
Assign each atom its cell conditional mean. On a zero-mass cell choose0 as this version, not as a uniquely determined conditional quotient. Weight these values by the original atom masses and compare E(E(X|G)) with E(X), where G is generated by the full partition. The exact result, retaining the requested order, is ([74/3, 74/3, 27, 27], 911/34, 911/34, Yes, conditional_expectation_constant_on_partition_cells).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-tower-identity · candidate
76. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/10, 2/10, 7/10, 0/1], aligned values X=[0, 1, 3, 99] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 23/10], [1, 0, null]], 23/10, 23/10, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 23/10], [1, 0, null]], 23/10, 23/10, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
77. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/11, 2/11, 8/11, 0/1], aligned values X=[1, 2, 4, 100] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 37/11], [1, 0, null]], 37/11, 37/11, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 37/11], [1, 0, null]], 37/11, 37/11, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
78. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/12, 2/12, 9/12, 0/1], aligned values X=[2, 3, 5, 101] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 53/12], [1, 0, null]], 53/12, 53/12, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 53/12], [1, 0, null]], 53/12, 53/12, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
79. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/13, 2/13, 10/13, 0/1], aligned values X=[3, 4, 6, 102] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 71/13], [1, 0, null]], 71/13, 71/13, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 71/13], [1, 0, null]], 71/13, 71/13, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
80. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/14, 2/14, 11/14, 0/1], aligned values X=[4, 5, 7, 103] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 13/2], [1, 0, null]], 13/2, 13/2, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 13/2], [1, 0, null]], 13/2, 13/2, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
81. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/15, 2/15, 12/15, 0/1], aligned values X=[5, 6, 8, 104] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 113/15], [1, 0, null]], 113/15, 113/15, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 113/15], [1, 0, null]], 113/15, 113/15, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
82. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/16, 2/16, 13/16, 0/1], aligned values X=[6, 7, 9, 105] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 137/16], [1, 0, null]], 137/16, 137/16, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 137/16], [1, 0, null]], 137/16, 137/16, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
83. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/17, 2/17, 14/17, 0/1], aligned values X=[7, 8, 10, 106] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 163/17], [1, 0, null]], 163/17, 163/17, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 163/17], [1, 0, null]], 163/17, 163/17, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
84. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/18, 2/18, 15/18, 0/1], aligned values X=[8, 9, 11, 107] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 191/18], [1, 0, null]], 191/18, 191/18, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 191/18], [1, 0, null]], 191/18, 191/18, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
85. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/19, 2/19, 16/19, 0/1], aligned values X=[9, 10, 12, 108] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 221/19], [1, 0, null]], 221/19, 221/19, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 221/19], [1, 0, null]], 221/19, 221/19, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
86. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/20, 2/20, 17/20, 0/1], aligned values X=[10, 11, 13, 109] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 253/20], [1, 0, null]], 253/20, 253/20, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 253/20], [1, 0, null]], 253/20, 253/20, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
87. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/21, 2/21, 18/21, 0/1], aligned values X=[11, 12, 14, 110] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 41/3], [1, 0, null]], 41/3, 41/3, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
88. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/22, 2/22, 19/22, 0/1], aligned values X=[12, 13, 15, 111] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 323/22], [1, 0, null]], 323/22, 323/22, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 323/22], [1, 0, null]], 323/22, 323/22, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
89. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 361/23], [1, 0, null]], 361/23, 361/23, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 361/23], [1, 0, null]], 361/23, 361/23, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
90. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/24, 2/24, 21/24, 0/1], aligned values X=[14, 15, 17, 113] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 401/24], [1, 0, null]], 401/24, 401/24, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 401/24], [1, 0, null]], 401/24, 401/24, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
91. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/25, 2/25, 22/25, 0/1], aligned values X=[15, 16, 18, 114] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 443/25], [1, 0, null]], 443/25, 443/25, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 443/25], [1, 0, null]], 443/25, 443/25, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
92. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/26, 2/26, 23/26, 0/1], aligned values X=[16, 17, 19, 115] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 487/26], [1, 0, null]], 487/26, 487/26, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 487/26], [1, 0, null]], 487/26, 487/26, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
93. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/27, 2/27, 24/27, 0/1], aligned values X=[17, 18, 20, 116] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 533/27], [1, 0, null]], 533/27, 533/27, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 533/27], [1, 0, null]], 533/27, 533/27, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
94. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/28, 2/28, 25/28, 0/1], aligned values X=[18, 19, 21, 117] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 83/4], [1, 0, null]], 83/4, 83/4, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 83/4], [1, 0, null]], 83/4, 83/4, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
95. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/29, 2/29, 26/29, 0/1], aligned values X=[19, 20, 22, 118] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 631/29], [1, 0, null]], 631/29, 631/29, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 631/29], [1, 0, null]], 631/29, 631/29, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
96. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/30, 2/30, 27/30, 0/1], aligned values X=[20, 21, 23, 119] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 683/30], [1, 0, null]], 683/30, 683/30, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 683/30], [1, 0, null]], 683/30, 683/30, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
97. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/31, 2/31, 28/31, 0/1], aligned values X=[21, 22, 24, 120] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 737/31], [1, 0, null]], 737/31, 737/31, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 737/31], [1, 0, null]], 737/31, 737/31, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
98. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/32, 2/32, 29/32, 0/1], aligned values X=[22, 23, 25, 121] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 793/32], [1, 0, null]], 793/32, 793/32, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 793/32], [1, 0, null]], 793/32, 793/32, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
99. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/33, 2/33, 30/33, 0/1], aligned values X=[23, 24, 26, 122] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 851/33], [1, 0, null]], 851/33, 851/33, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 851/33], [1, 0, null]], 851/33, 851/33, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
100. Null Cells Retain Undefined Quotients and Zero Weighted Contributions
Use the complete normalized finite law masses=[1/34, 2/34, 31/34, 0/1], aligned values X=[24, 25, 27, 123] and complete disjoint partition=[[0, 1, 2], [3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (cell_rows, positive_cell_weighted_sum, mean, equal, null_cell_division_claim, boundary).
cell_rows are [cell index,cell mass,conditional mean or null]. null preserves undefined conditional division; the cell has zero weighted contribution.
View answer and explanation
([[0, 1, 911/34], [1, 0, null]], 911/34, 911/34, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero)
Require at least one zero-mass cell. Each row is [cell index,cell mass,conditional mean or null]. Do not divide by zero or replace null by an asserted quotient. Sum only positive-cell weighted means; null cells contribute0. The final fixed No refuses null-cell division, not the tower identity. The exact result, retaining the requested order, is ([[0, 1, 911/34], [1, 0, null]], 911/34, 911/34, Yes, No, zero_mass_conditional_mean_not_divided_and_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total expectation · qf-mod40-totalexp-zero-mass-boundary · candidate
Canonical question JSONL · English JSONL · Validation report