Question 14137: 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=[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].

• G001 — 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].

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(20/49, 15/49, 8/49, 6/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[20, 49], [15, 49], [8, 49], [6, 49]]

normalized masses=[20/49, 15/49, 8/49, 6/49];nonnegative=Yes;total=1

same outcome indices=[0, 1, 2, 3];X values=[-1, 1, 13, 1];block labels=[0, 0, 1, 1]

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

block0 mass := sum(20/49, 15/49)

block0 mass = 5/7

block0 atom0 weighted X := mul(20/49, -1)

block0 atom0 weighted X = -20/49

block0 atom1 weighted X := mul(15/49, 1)

block0 atom1 weighted X = 15/49

block0 original weighted integral := sum(-20/49, 15/49)

block0 original weighted integral = -5/49

block0 positive-mass conditional mean := div(-5/49, 5/7)

block0 positive-mass conditional mean = -1/7

block label=0;indices=[0, 1];positive mass=5/7;weighted integral=-5/49

block-constant conditional value=-1/7

block1 mass := sum(8/49, 6/49)

block1 mass = 2/7

block1 atom2 weighted X := mul(8/49, 13)

block1 atom2 weighted X = 104/49

block1 atom3 weighted X := mul(6/49, 1)

block1 atom3 weighted X = 6/49

block1 original weighted integral := sum(104/49, 6/49)

block1 original weighted integral = 110/49

block1 positive-mass conditional mean := div(110/49, 2/7)

block1 positive-mass conditional mean = 55/7

block label=1;indices=[2, 3];positive mass=2/7;weighted integral=110/49

block-constant conditional value=55/7

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], 5/7, -1/7], [1, [2, 3], 2/7, 55/7]], [-1/7, -1/7, 55/7, 55/7])

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.