Nonempty Partition Cells, Exact Within-Between Variance and Null Weights
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batch-20261014-advance-20261110-r04-law-of-total-variance-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.
Field order uses a source-bound presentation supplement; canonical JSONL and mathematical answers are unchanged.
Field-order and source-binding data
1. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/10], [1, [2, 3], 7/10]], Yes, 23/10, 121/100)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/10], [1, [2, 3], 7/10]], Yes, 23/10, 121/100).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
2. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/11], [1, [2, 3], 8/11]], Yes, 37/11, 138/121)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/11], [1, [2, 3], 8/11]], Yes, 37/11, 138/121).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
3. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/4], [1, [2, 3], 3/4]], Yes, 53/12, 155/144)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/4], [1, [2, 3], 3/4]], Yes, 53/12, 155/144).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
4. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/13], [1, [2, 3], 10/13]], Yes, 71/13, 172/169)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/13], [1, [2, 3], 10/13]], Yes, 71/13, 172/169).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
5. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/14], [1, [2, 3], 11/14]], Yes, 13/2, 27/28)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/14], [1, [2, 3], 11/14]], Yes, 13/2, 27/28).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
6. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/5], [1, [2, 3], 4/5]], Yes, 113/15, 206/225)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/5], [1, [2, 3], 4/5]], Yes, 113/15, 206/225).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
7. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/16], [1, [2, 3], 13/16]], Yes, 137/16, 223/256)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/16], [1, [2, 3], 13/16]], Yes, 137/16, 223/256).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
8. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/17], [1, [2, 3], 14/17]], Yes, 163/17, 240/289)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/17], [1, [2, 3], 14/17]], Yes, 163/17, 240/289).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
9. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/6], [1, [2, 3], 5/6]], Yes, 191/18, 257/324)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/6], [1, [2, 3], 5/6]], Yes, 191/18, 257/324).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
10. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/19], [1, [2, 3], 16/19]], Yes, 221/19, 274/361)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/19], [1, [2, 3], 16/19]], Yes, 221/19, 274/361).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
11. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/20], [1, [2, 3], 17/20]], Yes, 253/20, 291/400)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/20], [1, [2, 3], 17/20]], Yes, 253/20, 291/400).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
12. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/7], [1, [2, 3], 6/7]], Yes, 41/3, 44/63)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/7], [1, [2, 3], 6/7]], Yes, 41/3, 44/63).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
13. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/22], [1, [2, 3], 19/22]], Yes, 323/22, 325/484)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/22], [1, [2, 3], 19/22]], Yes, 323/22, 325/484).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
14. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/23], [1, [2, 3], 20/23]], Yes, 361/23, 342/529)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/23], [1, [2, 3], 20/23]], Yes, 361/23, 342/529).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
15. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/8], [1, [2, 3], 7/8]], Yes, 401/24, 359/576)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/8], [1, [2, 3], 7/8]], Yes, 401/24, 359/576).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
16. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/25], [1, [2, 3], 22/25]], Yes, 443/25, 376/625)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/25], [1, [2, 3], 22/25]], Yes, 443/25, 376/625).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
17. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/26], [1, [2, 3], 23/26]], Yes, 487/26, 393/676)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/26], [1, [2, 3], 23/26]], Yes, 487/26, 393/676).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
18. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/9], [1, [2, 3], 8/9]], Yes, 533/27, 410/729)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/9], [1, [2, 3], 8/9]], Yes, 533/27, 410/729).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
19. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/28], [1, [2, 3], 25/28]], Yes, 83/4, 61/112)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/28], [1, [2, 3], 25/28]], Yes, 83/4, 61/112).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
20. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/29], [1, [2, 3], 26/29]], Yes, 631/29, 444/841)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/29], [1, [2, 3], 26/29]], Yes, 631/29, 444/841).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
21. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/10], [1, [2, 3], 9/10]], Yes, 683/30, 461/900)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/10], [1, [2, 3], 9/10]], Yes, 683/30, 461/900).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
22. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/31], [1, [2, 3], 28/31]], Yes, 737/31, 478/961)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/31], [1, [2, 3], 28/31]], Yes, 737/31, 478/961).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
23. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], Yes, 793/32, 495/1024)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/32], [1, [2, 3], 29/32]], Yes, 793/32, 495/1024).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
24. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 1/11], [1, [2, 3], 10/11]], Yes, 851/33, 512/1089)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 1/11], [1, [2, 3], 10/11]], Yes, 851/33, 512/1089).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
25. Nonempty Cells May Have Zero Mass without Becoming Empty Events
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (partition, cell_mass_rows, complete_disjoint_certificate, mean, variance).
View answer and explanation
([[0, 1], [2, 3]], [[0, [0, 1], 3/34], [1, [2, 3], 31/34]], Yes, 911/34, 529/1156)
Report cells and [cell index,cell atoms,cell mass], then the complete disjoint certificate, global mean and global variance. Zero-mass atoms remain part of the carrier and their containing nonempty cell. The exact result, retaining the requested order, is ([[0, 1], [2, 3]], [[0, [0, 1], 3/34], [1, [2, 3], 31/34]], Yes, 911/34, 529/1156).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-partition-qualification · candidate
26. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/10, 2/3, 2/9], [1, [2, 3], 7/10, 3, 0]], 1/15, 343/300, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/10, 2/3, 2/9], [1, [2, 3], 7/10, 3, 0]], 1/15, 343/300, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
27. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/11, 5/3, 2/9], [1, [2, 3], 8/11, 4, 0]], 2/33, 392/363, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/11, 5/3, 2/9], [1, [2, 3], 8/11, 4, 0]], 2/33, 392/363, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
28. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/4, 8/3, 2/9], [1, [2, 3], 3/4, 5, 0]], 1/18, 49/48, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/4, 8/3, 2/9], [1, [2, 3], 3/4, 5, 0]], 1/18, 49/48, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
29. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/13, 11/3, 2/9], [1, [2, 3], 10/13, 6, 0]], 2/39, 490/507, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/13, 11/3, 2/9], [1, [2, 3], 10/13, 6, 0]], 2/39, 490/507, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
30. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/14, 14/3, 2/9], [1, [2, 3], 11/14, 7, 0]], 1/21, 11/12, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/14, 14/3, 2/9], [1, [2, 3], 11/14, 7, 0]], 1/21, 11/12, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
31. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/5, 17/3, 2/9], [1, [2, 3], 4/5, 8, 0]], 2/45, 196/225, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/5, 17/3, 2/9], [1, [2, 3], 4/5, 8, 0]], 2/45, 196/225, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
32. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/16, 20/3, 2/9], [1, [2, 3], 13/16, 9, 0]], 1/24, 637/768, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/16, 20/3, 2/9], [1, [2, 3], 13/16, 9, 0]], 1/24, 637/768, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
33. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/17, 23/3, 2/9], [1, [2, 3], 14/17, 10, 0]], 2/51, 686/867, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/17, 23/3, 2/9], [1, [2, 3], 14/17, 10, 0]], 2/51, 686/867, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
34. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/6, 26/3, 2/9], [1, [2, 3], 5/6, 11, 0]], 1/27, 245/324, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/6, 26/3, 2/9], [1, [2, 3], 5/6, 11, 0]], 1/27, 245/324, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
35. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/19, 29/3, 2/9], [1, [2, 3], 16/19, 12, 0]], 2/57, 784/1083, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/19, 29/3, 2/9], [1, [2, 3], 16/19, 12, 0]], 2/57, 784/1083, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
36. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/20, 32/3, 2/9], [1, [2, 3], 17/20, 13, 0]], 1/30, 833/1200, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/20, 32/3, 2/9], [1, [2, 3], 17/20, 13, 0]], 1/30, 833/1200, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
37. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/7, 35/3, 2/9], [1, [2, 3], 6/7, 14, 0]], 2/63, 2/3, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/7, 35/3, 2/9], [1, [2, 3], 6/7, 14, 0]], 2/63, 2/3, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
38. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/22, 38/3, 2/9], [1, [2, 3], 19/22, 15, 0]], 1/33, 931/1452, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/22, 38/3, 2/9], [1, [2, 3], 19/22, 15, 0]], 1/33, 931/1452, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
39. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/23, 41/3, 2/9], [1, [2, 3], 20/23, 16, 0]], 2/69, 980/1587, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/23, 41/3, 2/9], [1, [2, 3], 20/23, 16, 0]], 2/69, 980/1587, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
40. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/8, 44/3, 2/9], [1, [2, 3], 7/8, 17, 0]], 1/36, 343/576, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/8, 44/3, 2/9], [1, [2, 3], 7/8, 17, 0]], 1/36, 343/576, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
41. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/25, 47/3, 2/9], [1, [2, 3], 22/25, 18, 0]], 2/75, 1078/1875, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/25, 47/3, 2/9], [1, [2, 3], 22/25, 18, 0]], 2/75, 1078/1875, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
42. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/26, 50/3, 2/9], [1, [2, 3], 23/26, 19, 0]], 1/39, 1127/2028, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/26, 50/3, 2/9], [1, [2, 3], 23/26, 19, 0]], 1/39, 1127/2028, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
43. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/9, 53/3, 2/9], [1, [2, 3], 8/9, 20, 0]], 2/81, 392/729, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/9, 53/3, 2/9], [1, [2, 3], 8/9, 20, 0]], 2/81, 392/729, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
44. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/28, 56/3, 2/9], [1, [2, 3], 25/28, 21, 0]], 1/42, 25/48, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/28, 56/3, 2/9], [1, [2, 3], 25/28, 21, 0]], 1/42, 25/48, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
45. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/29, 59/3, 2/9], [1, [2, 3], 26/29, 22, 0]], 2/87, 1274/2523, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/29, 59/3, 2/9], [1, [2, 3], 26/29, 22, 0]], 2/87, 1274/2523, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
46. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/10, 62/3, 2/9], [1, [2, 3], 9/10, 23, 0]], 1/45, 49/100, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/10, 62/3, 2/9], [1, [2, 3], 9/10, 23, 0]], 1/45, 49/100, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
47. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/31, 65/3, 2/9], [1, [2, 3], 28/31, 24, 0]], 2/93, 1372/2883, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/31, 65/3, 2/9], [1, [2, 3], 28/31, 24, 0]], 2/93, 1372/2883, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
48. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/32, 68/3, 2/9], [1, [2, 3], 29/32, 25, 0]], 1/48, 1421/3072, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/32, 68/3, 2/9], [1, [2, 3], 29/32, 25, 0]], 1/48, 1421/3072, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
49. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 1/11, 71/3, 2/9], [1, [2, 3], 10/11, 26, 0]], 2/99, 490/1089, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 1/11, 71/3, 2/9], [1, [2, 3], 10/11, 26, 0]], 2/99, 490/1089, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
50. Average Cell Variances and Vary Cell Means under the Same Law
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, certificate).
View answer and explanation
([[0, [0, 1], 3/34, 74/3, 2/9], [1, [2, 3], 31/34, 27, 0]], 1/51, 1519/3468, same_law_same_partition_within_and_between_terms)
Each row is [index,cell atoms,mass,conditional mean or null,conditional variance or null]. The within term is the mass-weighted cell variance; the between term is the mass-weighted squared deviation of cell means from the global mean. Use identical law and partition for both; null cells have zero outer weight. The exact result, retaining the requested order, is ([[0, [0, 1], 3/34, 74/3, 2/9], [1, [2, 3], 31/34, 27, 0]], 1/51, 1519/3468, same_law_same_partition_within_and_between_terms).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-within-between-components · candidate
51. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(121/100, 1/15, 343/300, 121/100, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (121/100, 1/15, 343/300, 121/100, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
52. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(138/121, 2/33, 392/363, 138/121, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (138/121, 2/33, 392/363, 138/121, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
53. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(155/144, 1/18, 49/48, 155/144, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (155/144, 1/18, 49/48, 155/144, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
54. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(172/169, 2/39, 490/507, 172/169, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (172/169, 2/39, 490/507, 172/169, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
55. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(27/28, 1/21, 11/12, 27/28, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (27/28, 1/21, 11/12, 27/28, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
56. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(206/225, 2/45, 196/225, 206/225, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (206/225, 2/45, 196/225, 206/225, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
57. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(223/256, 1/24, 637/768, 223/256, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (223/256, 1/24, 637/768, 223/256, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
58. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(240/289, 2/51, 686/867, 240/289, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (240/289, 2/51, 686/867, 240/289, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
59. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(257/324, 1/27, 245/324, 257/324, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (257/324, 1/27, 245/324, 257/324, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
60. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(274/361, 2/57, 784/1083, 274/361, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (274/361, 2/57, 784/1083, 274/361, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
61. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(291/400, 1/30, 833/1200, 291/400, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (291/400, 1/30, 833/1200, 291/400, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
62. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(44/63, 2/63, 2/3, 44/63, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (44/63, 2/63, 2/3, 44/63, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
63. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(325/484, 1/33, 931/1452, 325/484, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (325/484, 1/33, 931/1452, 325/484, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
64. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(342/529, 2/69, 980/1587, 342/529, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (342/529, 2/69, 980/1587, 342/529, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
65. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(359/576, 1/36, 343/576, 359/576, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (359/576, 1/36, 343/576, 359/576, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
66. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(376/625, 2/75, 1078/1875, 376/625, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (376/625, 2/75, 1078/1875, 376/625, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
67. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(393/676, 1/39, 1127/2028, 393/676, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (393/676, 1/39, 1127/2028, 393/676, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
68. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(410/729, 2/81, 392/729, 410/729, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (410/729, 2/81, 392/729, 410/729, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
69. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(61/112, 1/42, 25/48, 61/112, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (61/112, 1/42, 25/48, 61/112, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
70. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(444/841, 2/87, 1274/2523, 444/841, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (444/841, 2/87, 1274/2523, 444/841, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
71. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(461/900, 1/45, 49/100, 461/900, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (461/900, 1/45, 49/100, 461/900, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
72. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(478/961, 2/93, 1372/2883, 478/961, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (478/961, 2/93, 1372/2883, 478/961, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
73. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(495/1024, 1/48, 1421/3072, 495/1024, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (495/1024, 1/48, 1421/3072, 495/1024, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
74. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(512/1089, 2/99, 490/1089, 512/1089, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (512/1089, 2/99, 490/1089, 512/1089, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
75. Both Within and Between Terms Are Needed for Total Variance
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (variance, within, between, sum, equal).
View answer and explanation
(529/1156, 1/51, 1519/3468, 529/1156, Yes)
Compare Var(X), E[Var(X|G)], Var(E[X|G]) and their sum with exact rational arithmetic. Do not omit a term or interchange their meanings. No independence of partition events is assumed. The exact result, retaining the requested order, is (529/1156, 1/51, 1519/3468, 529/1156, Yes).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-full-identity · candidate
76. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 23/10, 121/100], [1, [3], 0, null, null]], 121/100, 0, 121/100, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 23/10, 121/100], [1, [3], 0, null, null]], 121/100, 0, 121/100, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
77. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 37/11, 138/121], [1, [3], 0, null, null]], 138/121, 0, 138/121, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 37/11, 138/121], [1, [3], 0, null, null]], 138/121, 0, 138/121, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
78. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 53/12, 155/144], [1, [3], 0, null, null]], 155/144, 0, 155/144, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 53/12, 155/144], [1, [3], 0, null, null]], 155/144, 0, 155/144, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
79. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 71/13, 172/169], [1, [3], 0, null, null]], 172/169, 0, 172/169, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 71/13, 172/169], [1, [3], 0, null, null]], 172/169, 0, 172/169, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
80. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 13/2, 27/28], [1, [3], 0, null, null]], 27/28, 0, 27/28, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 13/2, 27/28], [1, [3], 0, null, null]], 27/28, 0, 27/28, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
81. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 113/15, 206/225], [1, [3], 0, null, null]], 206/225, 0, 206/225, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 113/15, 206/225], [1, [3], 0, null, null]], 206/225, 0, 206/225, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
82. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 137/16, 223/256], [1, [3], 0, null, null]], 223/256, 0, 223/256, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 137/16, 223/256], [1, [3], 0, null, null]], 223/256, 0, 223/256, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
83. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 163/17, 240/289], [1, [3], 0, null, null]], 240/289, 0, 240/289, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 163/17, 240/289], [1, [3], 0, null, null]], 240/289, 0, 240/289, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
84. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 191/18, 257/324], [1, [3], 0, null, null]], 257/324, 0, 257/324, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 191/18, 257/324], [1, [3], 0, null, null]], 257/324, 0, 257/324, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
85. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 221/19, 274/361], [1, [3], 0, null, null]], 274/361, 0, 274/361, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 221/19, 274/361], [1, [3], 0, null, null]], 274/361, 0, 274/361, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
86. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 253/20, 291/400], [1, [3], 0, null, null]], 291/400, 0, 291/400, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 253/20, 291/400], [1, [3], 0, null, null]], 291/400, 0, 291/400, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
87. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 41/3, 44/63], [1, [3], 0, null, null]], 44/63, 0, 44/63, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 41/3, 44/63], [1, [3], 0, null, null]], 44/63, 0, 44/63, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
88. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 323/22, 325/484], [1, [3], 0, null, null]], 325/484, 0, 325/484, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 323/22, 325/484], [1, [3], 0, null, null]], 325/484, 0, 325/484, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
89. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 361/23, 342/529], [1, [3], 0, null, null]], 342/529, 0, 342/529, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 361/23, 342/529], [1, [3], 0, null, null]], 342/529, 0, 342/529, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
90. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 401/24, 359/576], [1, [3], 0, null, null]], 359/576, 0, 359/576, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 401/24, 359/576], [1, [3], 0, null, null]], 359/576, 0, 359/576, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
91. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 443/25, 376/625], [1, [3], 0, null, null]], 376/625, 0, 376/625, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 443/25, 376/625], [1, [3], 0, null, null]], 376/625, 0, 376/625, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
92. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 487/26, 393/676], [1, [3], 0, null, null]], 393/676, 0, 393/676, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 487/26, 393/676], [1, [3], 0, null, null]], 393/676, 0, 393/676, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
93. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 533/27, 410/729], [1, [3], 0, null, null]], 410/729, 0, 410/729, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 533/27, 410/729], [1, [3], 0, null, null]], 410/729, 0, 410/729, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
94. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 83/4, 61/112], [1, [3], 0, null, null]], 61/112, 0, 61/112, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 83/4, 61/112], [1, [3], 0, null, null]], 61/112, 0, 61/112, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
95. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 631/29, 444/841], [1, [3], 0, null, null]], 444/841, 0, 444/841, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 631/29, 444/841], [1, [3], 0, null, null]], 444/841, 0, 444/841, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
96. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 683/30, 461/900], [1, [3], 0, null, null]], 461/900, 0, 461/900, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 683/30, 461/900], [1, [3], 0, null, null]], 461/900, 0, 461/900, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
97. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 737/31, 478/961], [1, [3], 0, null, null]], 478/961, 0, 478/961, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 737/31, 478/961], [1, [3], 0, null, null]], 478/961, 0, 478/961, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
98. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 793/32, 495/1024], [1, [3], 0, null, null]], 495/1024, 0, 495/1024, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 793/32, 495/1024], [1, [3], 0, null, null]], 495/1024, 0, 495/1024, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
99. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 851/33, 512/1089], [1, [3], 0, null, null]], 512/1089, 0, 512/1089, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 851/33, 512/1089], [1, [3], 0, null, null]], 512/1089, 0, 512/1089, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
100. Undefined Null-Cell Mean and Variance Have Zero Outer Contribution
Use the same 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]]. Every cell is nonempty even if all its atoms have zero mass; cells cover the whole carrier. G is generated by this partition. Conditional division requires positive cell mass. No independence premise is required. Give (cell_rows, within, between, variance, equal, null_cell_division_claim, boundary).
cell_rows are [index,nonempty cell,mass,mean or null,variance or null]. A zero-mass cell is not empty; its contribution to both outer weighted terms is zero.
View answer and explanation
([[0, [0, 1, 2], 1, 911/34, 529/1156], [1, [3], 0, null, null]], 529/1156, 0, 529/1156, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero)
Require a zero-mass nonempty cell. Keep null for both conditional quotients, never divide by zero, and retain both complete within and between terms. The fixed No withholds null-cell division, not the total-variance identity. The exact result, retaining the requested order, is ([[0, [0, 1, 2], 1, 911/34, 529/1156], [1, [3], 0, null, null]], 529/1156, 0, 529/1156, Yes, No, zero_mass_cell_not_divided_and_outer_weighted_contribution_zero).
Reviewed step-by-step derivation
Law of total variance · qf-mod41-totalvar-zero-mass-boundary · candidate
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