Question 14131: Conditional Means Are Blockwise and Only Almost-Sure on Null Blocks

Original question and givens

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].

• G001 — 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].

Definition and scope

Use the original question domain, assumptions and cited MKO definitions. This item has second-context mathematical and bilingual review, with the model, answer and every arithmetic step independently recomputed from the original parameters; the source question remains a candidate.

• mko-conditional-expectation

Step-by-step derivation

conditional model whole mass sum := sum(30/49, 12/49, 5/49, 2/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[30, 49], [12, 49], [5, 49], [2, 49]]

normalized masses=[30/49, 12/49, 5/49, 2/49];nonnegative=Yes;total=1

same outcome indices=[0, 1, 2, 3];X values=[-7, 2, 7, 7];block labels=[0, 0, 1, 1]

distinct conditioning labels in increasing order=[0, 1]

block0 mass := sum(30/49, 12/49)

block0 mass = 6/7

block0 atom0 weighted X := mul(30/49, -7)

block0 atom0 weighted X = -30/7

block0 atom1 weighted X := mul(12/49, 2)

block0 atom1 weighted X = 24/49

block0 original weighted integral := sum(-30/7, 24/49)

block0 original weighted integral = -186/49

block0 positive-mass conditional mean := div(-186/49, 6/7)

block0 positive-mass conditional mean = -31/7

block label=0;indices=[0, 1];positive mass=6/7;weighted integral=-186/49

block-constant conditional value=-31/7

block1 mass := sum(5/49, 2/49)

block1 mass = 1/7

block1 atom2 weighted X := mul(5/49, 7)

block1 atom2 weighted X = 5/7

block1 atom3 weighted X := mul(2/49, 7)

block1 atom3 weighted X = 2/7

block1 original weighted integral := sum(5/7, 2/7)

block1 original weighted integral = 1

block1 positive-mass conditional mean := div(1, 1/7)

block1 positive-mass conditional mean = 7

block label=1;indices=[2, 3];positive mass=1/7;weighted integral=1

block-constant conditional value=7

Reverse checks

• S015 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S014 · premise_and_result_audit — Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute block1 original weighted integral with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute block1 atom3 weighted X with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute block1 atom2 weighted X with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute block1 mass with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute block0 original weighted integral with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute block0 atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute block0 atom0 weighted X with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute block0 mass with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S002 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S001 · premise_and_result_audit — Recompute conditional model whole mass sum with exact rational arithmetic, not rounded decimals.

Result

([[0, [0, 1], 6/7, -31/7], [1, [2, 3], 1/7, 7]], [-31/7, -31/7, 7, 7])

The full substitutions and premise checks give the original answer; source identity,format and status are unchanged.

This question review and source binding

Review date: 2026-10-26

Original questions, answers, step expressions, numeric ASTs and historical records are retained. Source candidate labels remain historical and are not overwritten as formal proof.