Question 14126: 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=[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].

• G001 — 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].

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(36/49, 6/49, 6/49, 1/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[36, 49], [6, 49], [6, 49], [1, 49]]

normalized masses=[36/49, 6/49, 6/49, 1/49];nonnegative=Yes;total=1

same outcome indices=[0, 1, 2, 3];X values=[-12, -3, 2, 12];block labels=[0, 1, 1, 2]

distinct conditioning labels in increasing order=[0, 1, 2]

block0 mass := sum(36/49)

block0 mass = 36/49

block0 atom0 weighted X := mul(36/49, -12)

block0 atom0 weighted X = -432/49

block0 original weighted integral := sum(-432/49)

block0 original weighted integral = -432/49

block0 positive-mass conditional mean := div(-432/49, 36/49)

block0 positive-mass conditional mean = -12

block label=0;indices=[0];positive mass=36/49;weighted integral=-432/49

block-constant conditional value=-12

block1 mass := sum(6/49, 6/49)

block1 mass = 12/49

block1 atom1 weighted X := mul(6/49, -3)

block1 atom1 weighted X = -18/49

block1 atom2 weighted X := mul(6/49, 2)

block1 atom2 weighted X = 12/49

block1 original weighted integral := sum(-18/49, 12/49)

block1 original weighted integral = -6/49

block1 positive-mass conditional mean := div(-6/49, 12/49)

block1 positive-mass conditional mean = -1/2

block label=1;indices=[1, 2];positive mass=12/49;weighted integral=-6/49

block-constant conditional value=-1/2

block2 mass := sum(1/49)

block2 mass = 1/49

block2 atom3 weighted X := mul(1/49, 12)

block2 atom3 weighted X = 12/49

block2 original weighted integral := sum(12/49)

block2 original weighted integral = 12/49

block2 positive-mass conditional mean := div(12/49, 1/49)

block2 positive-mass conditional mean = 12

block label=2;indices=[3];positive mass=1/49;weighted integral=12/49

block-constant conditional value=12

Reverse checks

• S019 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S018 · premise_and_result_audit — Recompute block2 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute block2 original weighted integral with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute block2 atom3 weighted X with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute block2 mass with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S013 · premise_and_result_audit — Recompute block1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute block1 original weighted integral 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 atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute block1 mass with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S007 · premise_and_result_audit — Recompute block0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute block0 original weighted integral 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], 36/49, -12], [1, [1, 2], 12/49, -1/2], [2, [3], 1/49, 12]], [-12, -1/2, -1/2, 12])

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.