Blockwise Conditional Means, Measurability, Integrals and Null-Block Boundaries
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batch-20261014-advance-20261029-r04-conditional-expectation-foundations · 2026-10-14 · 100 questions
This English edition is an AI candidate and does not yet have an OCME Atlas (GPT-6) Twin closure receipt.
1. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[36/49, 6/49, 6/49, 1/49], values X=[-12, -3, 2, 12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 36/49, -12], [1, [1, 2], 12/49, -1/2], [2, [3], 1/49, 12]], [-12, -1/2, -1/2, 12])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 36/49, -12], [1, [1, 2], 12/49, -1/2], [2, [3], 1/49, 12]], [-12, -1/2, -1/2, 12]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
2. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 5/49, 12/49, 2/49], values X=[-11, -2, 3, 11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 5/7, -68/7], [1, [2, 3], 2/7, 29/7]], [-68/7, -68/7, 29/7, 29/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 5/7, -68/7], [1, [2, 3], 2/7, 29/7]], [-68/7, -68/7, 29/7, 29/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
3. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 4/49, 18/49, 3/49], values X=[-10, -1, 4, 10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 24/49, -10], [1, [1, 2], 22/49, 34/11], [2, [3], 3/49, 10]], [-10, 34/11, 34/11, 10])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 24/49, -10], [1, [1, 2], 22/49, 34/11], [2, [3], 3/49, 10]], [-10, 34/11, 34/11, 10]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
4. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 3/49, 24/49, 4/49], values X=[-9, 0, 5, 9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 3/7, -54/7], [1, [2, 3], 4/7, 39/7]], [-54/7, -54/7, 39/7, 39/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 3/7, -54/7], [1, [2, 3], 4/7, 39/7]], [-54/7, -54/7, 39/7, 39/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
5. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 2/49, 30/49, 5/49], values X=[-8, 1, 6, 8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 12/49, -8], [1, [1, 2], 32/49, 91/16], [2, [3], 5/49, 8]], [-8, 91/16, 91/16, 8])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 12/49, -8], [1, [1, 2], 32/49, 91/16], [2, [3], 5/49, 8]], [-8, 91/16, 91/16, 8]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
6. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 12/49, 5/49, 2/49], values X=[-7, 2, 7, 7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 6/7, -31/7], [1, [2, 3], 1/7, 7]], [-31/7, -31/7, 7, 7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 6/7, -31/7], [1, [2, 3], 1/7, 7]], [-31/7, -31/7, 7, 7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
7. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[25/49, 10/49, 10/49, 4/49], values X=[-6, 3, 8, 6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 25/49, -6], [1, [1, 2], 20/49, 11/2], [2, [3], 4/49, 6]], [-6, 11/2, 11/2, 6])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 25/49, -6], [1, [1, 2], 20/49, 11/2], [2, [3], 4/49, 6]], [-6, 11/2, 11/2, 6]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
8. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 8/49, 15/49, 6/49], values X=[-5, -3, 9, 5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 4/7, -31/7], [1, [2, 3], 3/7, 55/7]], [-31/7, -31/7, 55/7, 55/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 4/7, -31/7], [1, [2, 3], 3/7, 55/7]], [-31/7, -31/7, 55/7, 55/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
9. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 6/49, 20/49, 8/49], values X=[-4, -2, 10, 4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 15/49, -4], [1, [1, 2], 26/49, 94/13], [2, [3], 8/49, 4]], [-4, 94/13, 94/13, 4])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 15/49, -4], [1, [1, 2], 26/49, 94/13], [2, [3], 8/49, 4]], [-4, 94/13, 94/13, 4]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
10. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 4/49, 25/49, 10/49], values X=[-3, -1, 11, 3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 2/7, -17/7], [1, [2, 3], 5/7, 61/7]], [-17/7, -17/7, 61/7, 61/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 2/7, -17/7], [1, [2, 3], 5/7, 61/7]], [-17/7, -17/7, 61/7, 61/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
11. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 18/49, 4/49, 3/49], values X=[-2, 0, 12, 2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 24/49, -2], [1, [1, 2], 22/49, 24/11], [2, [3], 3/49, 2]], [-2, 24/11, 24/11, 2])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 24/49, -2], [1, [1, 2], 22/49, 24/11], [2, [3], 3/49, 2]], [-2, 24/11, 24/11, 2]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
12. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 15/49, 8/49, 6/49], values X=[-1, 1, 13, 1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 5/7, -1/7], [1, [2, 3], 2/7, 55/7]], [-1/7, -1/7, 55/7, 55/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 5/7, -1/7], [1, [2, 3], 2/7, 55/7]], [-1/7, -1/7, 55/7, 55/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
13. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[16/49, 12/49, 12/49, 9/49], values X=[0, 2, 14, 0], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 16/49, 0], [1, [1, 2], 24/49, 8], [2, [3], 9/49, 0]], [0, 8, 8, 0])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 16/49, 0], [1, [1, 2], 24/49, 8], [2, [3], 9/49, 0]], [0, 8, 8, 0]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
14. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 9/49, 16/49, 12/49], values X=[1, 3, 15, -1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 3/7, 13/7], [1, [2, 3], 4/7, 57/7]], [13/7, 13/7, 57/7, 57/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 3/7, 13/7], [1, [2, 3], 4/7, 57/7]], [13/7, 13/7, 57/7, 57/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
15. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 6/49, 20/49, 15/49], values X=[2, -3, 16, -2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 8/49, 2], [1, [1, 2], 26/49, 151/13], [2, [3], 15/49, -2]], [2, 151/13, 151/13, -2])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 8/49, 2], [1, [1, 2], 26/49, 151/13], [2, [3], 15/49, -2]], [2, 151/13, 151/13, -2]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
16. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 24/49, 3/49, 4/49], values X=[3, -2, 17, -3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 6/7, 1/7], [1, [2, 3], 1/7, 39/7]], [1/7, 1/7, 39/7, 39/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 6/7, 1/7], [1, [2, 3], 1/7, 39/7]], [1/7, 1/7, 39/7, 39/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
17. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 20/49, 6/49, 8/49], values X=[4, -1, 18, -4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 15/49, 4], [1, [1, 2], 26/49, 44/13], [2, [3], 8/49, -4]], [4, 44/13, 44/13, -4])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 15/49, 4], [1, [1, 2], 26/49, 44/13], [2, [3], 8/49, -4]], [4, 44/13, 44/13, -4]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
18. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 16/49, 9/49, 12/49], values X=[5, 0, 19, -5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 4/7, 15/7], [1, [2, 3], 3/7, 37/7]], [15/7, 15/7, 37/7, 37/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 4/7, 15/7], [1, [2, 3], 3/7, 37/7]], [15/7, 15/7, 37/7, 37/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
19. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[9/49, 12/49, 12/49, 16/49], values X=[6, 1, 20, -6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 9/49, 6], [1, [1, 2], 24/49, 21/2], [2, [3], 16/49, -6]], [6, 21/2, 21/2, -6])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 9/49, 6], [1, [1, 2], 24/49, 21/2], [2, [3], 16/49, -6]], [6, 21/2, 21/2, -6]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
20. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 8/49, 15/49, 20/49], values X=[7, 2, 21, -7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 2/7, 29/7], [1, [2, 3], 5/7, 5]], [29/7, 29/7, 5, 5])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 2/7, 29/7], [1, [2, 3], 5/7, 5]], [29/7, 29/7, 5, 5]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
21. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 30/49, 2/49, 5/49], values X=[8, 3, 22, -8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 12/49, 8], [1, [1, 2], 32/49, 67/16], [2, [3], 5/49, -8]], [8, 67/16, 67/16, -8])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 12/49, 8], [1, [1, 2], 32/49, 67/16], [2, [3], 5/49, -8]], [8, 67/16, 67/16, -8]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
22. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 25/49, 4/49, 10/49], values X=[9, -3, 23, -9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 5/7, 3/7], [1, [2, 3], 2/7, 1/7]], [3/7, 3/7, 1/7, 1/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 5/7, 3/7], [1, [2, 3], 2/7, 1/7]], [3/7, 3/7, 1/7, 1/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
23. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 20/49, 6/49, 15/49], values X=[10, -2, 24, -10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 8/49, 10], [1, [1, 2], 26/49, 4], [2, [3], 15/49, -10]], [10, 4, 4, -10])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 8/49, 10], [1, [1, 2], 26/49, 4], [2, [3], 15/49, -10]], [10, 4, 4, -10]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
24. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 15/49, 8/49, 20/49], values X=[11, -1, 25, -11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0, 1], 3/7, 17/7], [1, [2, 3], 4/7, -5/7]], [17/7, 17/7, -5/7, -5/7])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0, 1], 3/7, 17/7], [1, [2, 3], 4/7, -5/7]], [17/7, 17/7, -5/7, -5/7]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
25. Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[4/49, 10/49, 10/49, 25/49], values X=[12, 0, 26, -12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (block_summaries,conditional_values). Each summary is [label,indices,block_mass,block_mean_or_a.s._arbitrary].
View answer and explanation
([[0, [0], 4/49, 12], [1, [1, 2], 20/49, 13], [2, [3], 25/49, -12]], [12, 13, 13, -12])
Sum exact masses and mass-times-X within each complete block. Divide only for positive block mass and copy the resulting mean to all outcomes in that block. Return the null marker rather than an arbitrary forced point value. The exact result, retaining the requested order, is ([[0, [0], 4/49, 12], [1, [1, 2], 20/49, 13], [2, [3], 25/49, -12]], [12, 13, 13, -12]).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-values · candidate
26. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[36/49, 6/49, 6/49, 1/49], values X=[-12, -3, 2, 12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-12, -1/2, -1/2, 12], [[0, [0], [-12], Yes], [1, [1, 2], [-1/2], Yes], [2, [3], [12], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-12, -1/2, -1/2, 12], [[0, [0], [-12], Yes], [1, [1, 2], [-1/2], Yes], [2, [3], [12], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
27. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 5/49, 12/49, 2/49], values X=[-11, -2, 3, 11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-68/7, -68/7, 29/7, 29/7], [[0, [0, 1], [-68/7], Yes], [1, [2, 3], [29/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-68/7, -68/7, 29/7, 29/7], [[0, [0, 1], [-68/7], Yes], [1, [2, 3], [29/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
28. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 4/49, 18/49, 3/49], values X=[-10, -1, 4, 10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-10, 34/11, 34/11, 10], [[0, [0], [-10], Yes], [1, [1, 2], [34/11], Yes], [2, [3], [10], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-10, 34/11, 34/11, 10], [[0, [0], [-10], Yes], [1, [1, 2], [34/11], Yes], [2, [3], [10], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
29. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 3/49, 24/49, 4/49], values X=[-9, 0, 5, 9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-54/7, -54/7, 39/7, 39/7], [[0, [0, 1], [-54/7], Yes], [1, [2, 3], [39/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-54/7, -54/7, 39/7, 39/7], [[0, [0, 1], [-54/7], Yes], [1, [2, 3], [39/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
30. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 2/49, 30/49, 5/49], values X=[-8, 1, 6, 8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-8, 91/16, 91/16, 8], [[0, [0], [-8], Yes], [1, [1, 2], [91/16], Yes], [2, [3], [8], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-8, 91/16, 91/16, 8], [[0, [0], [-8], Yes], [1, [1, 2], [91/16], Yes], [2, [3], [8], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
31. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 12/49, 5/49, 2/49], values X=[-7, 2, 7, 7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-31/7, -31/7, 7, 7], [[0, [0, 1], [-31/7], Yes], [1, [2, 3], [7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-31/7, -31/7, 7, 7], [[0, [0, 1], [-31/7], Yes], [1, [2, 3], [7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
32. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[25/49, 10/49, 10/49, 4/49], values X=[-6, 3, 8, 6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-6, 11/2, 11/2, 6], [[0, [0], [-6], Yes], [1, [1, 2], [11/2], Yes], [2, [3], [6], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-6, 11/2, 11/2, 6], [[0, [0], [-6], Yes], [1, [1, 2], [11/2], Yes], [2, [3], [6], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
33. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 8/49, 15/49, 6/49], values X=[-5, -3, 9, 5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-31/7, -31/7, 55/7, 55/7], [[0, [0, 1], [-31/7], Yes], [1, [2, 3], [55/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-31/7, -31/7, 55/7, 55/7], [[0, [0, 1], [-31/7], Yes], [1, [2, 3], [55/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
34. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 6/49, 20/49, 8/49], values X=[-4, -2, 10, 4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-4, 94/13, 94/13, 4], [[0, [0], [-4], Yes], [1, [1, 2], [94/13], Yes], [2, [3], [4], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-4, 94/13, 94/13, 4], [[0, [0], [-4], Yes], [1, [1, 2], [94/13], Yes], [2, [3], [4], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
35. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 4/49, 25/49, 10/49], values X=[-3, -1, 11, 3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-17/7, -17/7, 61/7, 61/7], [[0, [0, 1], [-17/7], Yes], [1, [2, 3], [61/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-17/7, -17/7, 61/7, 61/7], [[0, [0, 1], [-17/7], Yes], [1, [2, 3], [61/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
36. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 18/49, 4/49, 3/49], values X=[-2, 0, 12, 2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-2, 24/11, 24/11, 2], [[0, [0], [-2], Yes], [1, [1, 2], [24/11], Yes], [2, [3], [2], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-2, 24/11, 24/11, 2], [[0, [0], [-2], Yes], [1, [1, 2], [24/11], Yes], [2, [3], [2], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
37. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 15/49, 8/49, 6/49], values X=[-1, 1, 13, 1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([-1/7, -1/7, 55/7, 55/7], [[0, [0, 1], [-1/7], Yes], [1, [2, 3], [55/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([-1/7, -1/7, 55/7, 55/7], [[0, [0, 1], [-1/7], Yes], [1, [2, 3], [55/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
38. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[16/49, 12/49, 12/49, 9/49], values X=[0, 2, 14, 0], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([0, 8, 8, 0], [[0, [0], [0], Yes], [1, [1, 2], [8], Yes], [2, [3], [0], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([0, 8, 8, 0], [[0, [0], [0], Yes], [1, [1, 2], [8], Yes], [2, [3], [0], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
39. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 9/49, 16/49, 12/49], values X=[1, 3, 15, -1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([13/7, 13/7, 57/7, 57/7], [[0, [0, 1], [13/7], Yes], [1, [2, 3], [57/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([13/7, 13/7, 57/7, 57/7], [[0, [0, 1], [13/7], Yes], [1, [2, 3], [57/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
40. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 6/49, 20/49, 15/49], values X=[2, -3, 16, -2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([2, 151/13, 151/13, -2], [[0, [0], [2], Yes], [1, [1, 2], [151/13], Yes], [2, [3], [-2], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([2, 151/13, 151/13, -2], [[0, [0], [2], Yes], [1, [1, 2], [151/13], Yes], [2, [3], [-2], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
41. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 24/49, 3/49, 4/49], values X=[3, -2, 17, -3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([1/7, 1/7, 39/7, 39/7], [[0, [0, 1], [1/7], Yes], [1, [2, 3], [39/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([1/7, 1/7, 39/7, 39/7], [[0, [0, 1], [1/7], Yes], [1, [2, 3], [39/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
42. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 20/49, 6/49, 8/49], values X=[4, -1, 18, -4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([4, 44/13, 44/13, -4], [[0, [0], [4], Yes], [1, [1, 2], [44/13], Yes], [2, [3], [-4], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([4, 44/13, 44/13, -4], [[0, [0], [4], Yes], [1, [1, 2], [44/13], Yes], [2, [3], [-4], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
43. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 16/49, 9/49, 12/49], values X=[5, 0, 19, -5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([15/7, 15/7, 37/7, 37/7], [[0, [0, 1], [15/7], Yes], [1, [2, 3], [37/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([15/7, 15/7, 37/7, 37/7], [[0, [0, 1], [15/7], Yes], [1, [2, 3], [37/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
44. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[9/49, 12/49, 12/49, 16/49], values X=[6, 1, 20, -6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([6, 21/2, 21/2, -6], [[0, [0], [6], Yes], [1, [1, 2], [21/2], Yes], [2, [3], [-6], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([6, 21/2, 21/2, -6], [[0, [0], [6], Yes], [1, [1, 2], [21/2], Yes], [2, [3], [-6], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
45. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 8/49, 15/49, 20/49], values X=[7, 2, 21, -7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([29/7, 29/7, 5, 5], [[0, [0, 1], [29/7], Yes], [1, [2, 3], [5], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([29/7, 29/7, 5, 5], [[0, [0, 1], [29/7], Yes], [1, [2, 3], [5], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
46. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 30/49, 2/49, 5/49], values X=[8, 3, 22, -8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([8, 67/16, 67/16, -8], [[0, [0], [8], Yes], [1, [1, 2], [67/16], Yes], [2, [3], [-8], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([8, 67/16, 67/16, -8], [[0, [0], [8], Yes], [1, [1, 2], [67/16], Yes], [2, [3], [-8], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
47. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 25/49, 4/49, 10/49], values X=[9, -3, 23, -9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([3/7, 3/7, 1/7, 1/7], [[0, [0, 1], [3/7], Yes], [1, [2, 3], [1/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([3/7, 3/7, 1/7, 1/7], [[0, [0, 1], [3/7], Yes], [1, [2, 3], [1/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
48. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 20/49, 6/49, 15/49], values X=[10, -2, 24, -10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([10, 4, 4, -10], [[0, [0], [10], Yes], [1, [1, 2], [4], Yes], [2, [3], [-10], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([10, 4, 4, -10], [[0, [0], [10], Yes], [1, [1, 2], [4], Yes], [2, [3], [-10], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
49. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 15/49, 8/49, 20/49], values X=[11, -1, 25, -11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([17/7, 17/7, -5/7, -5/7], [[0, [0, 1], [17/7], Yes], [1, [2, 3], [-5/7], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([17/7, 17/7, -5/7, -5/7], [[0, [0, 1], [17/7], Yes], [1, [2, 3], [-5/7], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
50. Block-Constant Values Give Conditioning Measurability
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[4/49, 10/49, 10/49, 25/49], values X=[12, 0, 26, -12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (conditional_values,block_rows,measurable?). Each block row is [label,indices,distinct_conditional_values,constant?].
View answer and explanation
([12, 13, 13, -12], [[0, [0], [12], Yes], [1, [1, 2], [13], Yes], [2, [3], [-12], Yes]], Yes)
Construct the same blockwise conditional values and list each block's distinct values in first-occurrence order. The a.s._arbitrary marker records a common unspecified null-block value; it does not assert a unique version there. The exact result, retaining the requested order, is ([12, 13, 13, -12], [[0, [0], [12], Yes], [1, [1, 2], [13], Yes], [2, [3], [-12], Yes]], Yes).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-measurability · candidate
51. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[36/49, 6/49, 6/49, 1/49], values X=[-12, -3, 2, 12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], -432/49, -432/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], -432/49, -432/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
52. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 5/49, 12/49, 2/49], values X=[-11, -2, 3, 11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0, 1], -340/49, -340/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0, 1], -340/49, -340/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
53. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 4/49, 18/49, 3/49], values X=[-10, -1, 4, 10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [1, 2], 68/49, 68/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [1, 2], 68/49, 68/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
54. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 3/49, 24/49, 4/49], values X=[-9, 0, 5, 9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [2, 3], 156/49, 156/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [2, 3], 156/49, 156/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
55. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 2/49, 30/49, 5/49], values X=[-8, 1, 6, 8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], -96/49, -96/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], -96/49, -96/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
56. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 12/49, 5/49, 2/49], values X=[-7, 2, 7, 7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0, 1], -186/49, -186/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0, 1], -186/49, -186/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
57. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[25/49, 10/49, 10/49, 4/49], values X=[-6, 3, 8, 6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [1, 2], 110/49, 110/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [1, 2], 110/49, 110/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
58. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 8/49, 15/49, 6/49], values X=[-5, -3, 9, 5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [2, 3], 165/49, 165/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [2, 3], 165/49, 165/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
59. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 6/49, 20/49, 8/49], values X=[-4, -2, 10, 4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], -60/49, -60/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], -60/49, -60/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
60. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 4/49, 25/49, 10/49], values X=[-3, -1, 11, 3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0, 1], -34/49, -34/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0, 1], -34/49, -34/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
61. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 18/49, 4/49, 3/49], values X=[-2, 0, 12, 2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [1, 2], 48/49, 48/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [1, 2], 48/49, 48/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
62. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 15/49, 8/49, 6/49], values X=[-1, 1, 13, 1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [2, 3], 110/49, 110/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [2, 3], 110/49, 110/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
63. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[16/49, 12/49, 12/49, 9/49], values X=[0, 2, 14, 0], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], 0, 0, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], 0, 0, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
64. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 9/49, 16/49, 12/49], values X=[1, 3, 15, -1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0, 1], 39/49, 39/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0, 1], 39/49, 39/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
65. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 6/49, 20/49, 15/49], values X=[2, -3, 16, -2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [1, 2], 302/49, 302/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [1, 2], 302/49, 302/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
66. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 24/49, 3/49, 4/49], values X=[3, -2, 17, -3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [2, 3], 39/49, 39/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [2, 3], 39/49, 39/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
67. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 20/49, 6/49, 8/49], values X=[4, -1, 18, -4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], 60/49, 60/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], 60/49, 60/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
68. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 16/49, 9/49, 12/49], values X=[5, 0, 19, -5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0, 1], 60/49, 60/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0, 1], 60/49, 60/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
69. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[9/49, 12/49, 12/49, 16/49], values X=[6, 1, 20, -6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [1, 2], 36/7, 36/7, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [1, 2], 36/7, 36/7, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
70. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 8/49, 15/49, 20/49], values X=[7, 2, 21, -7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [2, 3], 25/7, 25/7, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [2, 3], 25/7, 25/7, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
71. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 30/49, 2/49, 5/49], values X=[8, 3, 22, -8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], 96/49, 96/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], 96/49, 96/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
72. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 25/49, 4/49, 10/49], values X=[9, -3, 23, -9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0, 1], 15/49, 15/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0, 1], 15/49, 15/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
73. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 20/49, 6/49, 15/49], values X=[10, -2, 24, -10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [1, 2], 104/49, 104/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [1, 2], 104/49, 104/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
74. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 15/49, 8/49, 20/49], values X=[11, -1, 25, -11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=1 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(1, [2, 3], -20/49, -20/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (1, [2, 3], -20/49, -20/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
75. Preserve Every Selected Conditioning-Block Integral
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[4/49, 10/49, 10/49, 25/49], values X=[12, 0, 26, -12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Selected block_label=0 must exist. Give (block_label,indices,original_integral,conditional_integral,equal?,boundary). Boundary is positive_block or null_block_a.s._arbitrary.
View answer and explanation
(0, [0], 48/49, 48/49, Yes, positive_block)
Compute the original weighted integral on this block and compare with block_mass*block_mean. Both integrals are zero on a null block without performing division. Checking total expectation alone would not establish every block obligation. The exact result, retaining the requested order, is (0, [0], 48/49, 48/49, Yes, positive_block).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-block-integral · candidate
76. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[36/49, 6/49, 6/49, 1/49], values X=[-12, -3, 2, 12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(-426/49, -426/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (-426/49, -426/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
77. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 5/49, 12/49, 2/49], values X=[-11, -2, 3, 11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(-282/49, -282/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (-282/49, -282/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
78. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 4/49, 18/49, 3/49], values X=[-10, -1, 4, 10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(-142/49, -142/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (-142/49, -142/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
79. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 3/49, 24/49, 4/49], values X=[-9, 0, 5, 9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(-6/49, -6/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (-6/49, -6/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
80. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 2/49, 30/49, 5/49], values X=[-8, 1, 6, 8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(18/7, 18/7, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (18/7, 18/7, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
81. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[30/49, 12/49, 5/49, 2/49], values X=[-7, 2, 7, 7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(-137/49, -137/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (-137/49, -137/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
82. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[25/49, 10/49, 10/49, 4/49], values X=[-6, 3, 8, 6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(-16/49, -16/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (-16/49, -16/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
83. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 8/49, 15/49, 6/49], values X=[-5, -3, 9, 5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(41/49, 41/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (41/49, 41/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
84. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 6/49, 20/49, 8/49], values X=[-4, -2, 10, 4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(160/49, 160/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (160/49, 160/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
85. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 4/49, 25/49, 10/49], values X=[-3, -1, 11, 3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(271/49, 271/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (271/49, 271/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
86. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[24/49, 18/49, 4/49, 3/49], values X=[-2, 0, 12, 2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(6/49, 6/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (6/49, 6/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
87. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[20/49, 15/49, 8/49, 6/49], values X=[-1, 1, 13, 1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(15/7, 15/7, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (15/7, 15/7, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
88. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[16/49, 12/49, 12/49, 9/49], values X=[0, 2, 14, 0], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(192/49, 192/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (192/49, 192/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
89. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 9/49, 16/49, 12/49], values X=[1, 3, 15, -1], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(267/49, 267/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (267/49, 267/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
90. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 6/49, 20/49, 15/49], values X=[2, -3, 16, -2], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(288/49, 288/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (288/49, 288/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
91. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[18/49, 24/49, 3/49, 4/49], values X=[3, -2, 17, -3], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(45/49, 45/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (45/49, 45/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
92. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[15/49, 20/49, 6/49, 8/49], values X=[4, -1, 18, -4], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(116/49, 116/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (116/49, 116/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
93. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 16/49, 9/49, 12/49], values X=[5, 0, 19, -5], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(171/49, 171/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (171/49, 171/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
94. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[9/49, 12/49, 12/49, 16/49], values X=[6, 1, 20, -6], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(30/7, 30/7, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (30/7, 30/7, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
95. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 8/49, 15/49, 20/49], values X=[7, 2, 21, -7], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(233/49, 233/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (233/49, 233/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
96. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[12/49, 30/49, 2/49, 5/49], values X=[8, 3, 22, -8], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(190/49, 190/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (190/49, 190/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
97. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[10/49, 25/49, 4/49, 10/49], values X=[9, -3, 23, -9], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(17/49, 17/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (17/49, 17/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
98. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[8/49, 20/49, 6/49, 15/49], values X=[10, -2, 24, -10], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(34/49, 34/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (34/49, 34/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
99. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[6/49, 15/49, 8/49, 20/49], values X=[11, -1, 25, -11], and conditioning-block labels=[0, 0, 1, 1]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(31/49, 31/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (31/49, 31/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
100. Total Expectation Is a Consequence, Not the Whole Definition
Use one complete finite probability space with outcome indices 0 through 3, aligned masses=[4/49, 10/49, 10/49, 25/49], values X=[12, 0, 26, -12], and conditioning-block labels=[0, 1, 1, 2]. Masses are nonnegative and total exactly 1. Block labels describe a partition of this same space; list block summaries in increasing label order and outcomes in original index order. Conditional expectation is constant on each positive-mass block, with its weighted mean, and preserves every block integral. A zero-mass block is only determined almost surely: return a.s._arbitrary, never divide by zero. Use exact reduced fractions with positive denominators, omitting /1 for integers. Give (E_X,E_conditional,equal?,tower_to_trivial_sigma_algebra).
View answer and explanation
(8/49, 8/49, Yes, tower_to_trivial_sigma_algebra)
Aggregate the original weighted block integrals and the conditional block integrals over the same whole probability space. The equality is the tower consequence to the trivial sigma-algebra; it does not replace block measurability and all block integral checks. The exact result, retaining the requested order, is (8/49, 8/49, Yes, tower_to_trivial_sigma_algebra).
Reviewed step-by-step derivation
Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate
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