# 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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0001/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0002/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0003/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0004/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0005/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0006/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0007/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0008/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0009/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0010/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0011/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0012/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0013/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0014/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0015/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0016/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0017/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0018/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0019/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0020/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0021/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0022/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0023/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0024/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0025/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0026/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0027/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0028/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0029/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0030/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0031/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0032/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0033/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0034/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0035/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0036/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0037/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0038/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0039/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0040/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0041/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0042/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0043/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0044/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0045/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0046/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0047/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0048/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0049/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0050/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0051/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0052/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0053/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0054/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0055/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0056/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0057/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0058/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0059/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0060/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0061/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0062/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0063/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0064/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0065/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0066/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0067/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0068/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0069/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0070/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0071/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0072/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0073/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0074/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0075/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0076/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0077/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0078/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0079/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0080/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0081/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0082/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0083/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0084/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0085/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0086/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0087/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0088/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0089/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0090/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0091/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0092/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0093/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0094/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0095/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0096/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0097/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0098/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0099/)

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](https://ocme.evemisslab.com/en/questions/q-20261014-advance-20261029-r04-conditional-expectation-0100/)

Finite conditional expectation · qf-mod29-condexp-total-expectation · candidate



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