Question 14114: Preserve Every Selected Conditioning-Block Integral

Original question and givens

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.

• G001 — 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.

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(12/49, 9/49, 16/49, 12/49)

conditional model whole mass sum = 1

conditional model:raw aligned masses=[[12, 49], [9, 49], [16, 49], [12, 49]]

normalized masses=[12/49, 9/49, 16/49, 12/49];nonnegative=Yes;total=1

same outcome indices=[0, 1, 2, 3];X values=[1, 3, 15, -1];block labels=[0, 0, 1, 1]

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

block0 mass := sum(12/49, 9/49)

block0 mass = 3/7

block0 atom0 weighted X := mul(12/49, 1)

block0 atom0 weighted X = 12/49

block0 atom1 weighted X := mul(9/49, 3)

block0 atom1 weighted X = 27/49

block0 original weighted integral := sum(12/49, 27/49)

block0 original weighted integral = 39/49

block0 positive-mass conditional mean := div(39/49, 3/7)

block0 positive-mass conditional mean = 13/7

block label=0;indices=[0, 1];positive mass=3/7;weighted integral=39/49

block-constant conditional value=13/7

block1 mass := sum(16/49, 12/49)

block1 mass = 4/7

block1 atom2 weighted X := mul(16/49, 15)

block1 atom2 weighted X = 240/49

block1 atom3 weighted X := mul(12/49, -1)

block1 atom3 weighted X = -12/49

block1 original weighted integral := sum(240/49, -12/49)

block1 original weighted integral = 228/49

block1 positive-mass conditional mean := div(228/49, 4/7)

block1 positive-mass conditional mean = 57/7

block label=1;indices=[2, 3];positive mass=4/7;weighted integral=228/49

block-constant conditional value=57/7

selected block block0 conditional integral := mul(3/7, 13/7)

selected block block0 conditional integral = 39/49

selected block=0;indices=[0, 1];original integral=39/49;conditional integral=39/49

equal=Yes;this is the selected block obligation,not only a global average

Reverse checks

• S017 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S016 · premise_and_result_audit — Recompute selected block block0 conditional integral with exact rational arithmetic, not rounded decimals.
• 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], 39/49, 39/49, Yes, positive_block)

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.