# Question 21964: A Cell-Constant Conditional Version Has the Same Expectation [Return to catalogue page 110](https://ocme.evemisslab.com/en/questions/page/110/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261109-r04-law-of-total-expectation-foundations/#q-20261014-advance-20261109-r04-law-of-total-expectation-0064) This teaching derivation has second-context AI mathematical and bilingual semantic review, limited to the model and assumptions stated in this question. It has no human review or new Lean proof and does not promote the original question or MKO to Canonical truth. R0042-21964 · q-20261014-advance-20261109-r04-law-of-total-expectation-0064 ## Original question and givens Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate). - G001 — Use the complete normalized finite law masses=[1/23, 2/23, 20/23, 0/1], aligned values X=[13, 14, 16, 112] and complete disjoint partition=[[0, 1], [2, 3]] covering every atom. Values are integrable in this finite model. Cell order and atom order are retained; independence is not required. Conditional quotients require strictly positive cell mass. Give (conditional_values, tower_expectation, mean, equal, certificate). ## 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-law-of-total-expectation](https://ocme.evemisslab.com/en/mko/mko-law-of-total-expectation/) ## Step-by-step derivation S001 · impliesBefore source operation=totalexp-tower-identity;original snapshot and mathematics retained After explicit ordered source output fields=[conditional_values, tower_expectation, mean, equal, certificate] Conditions: none Keep the original historical interface marker while listing ordered source fields without JavaScript.Tags are not additional evidence;fixed No markers retain their source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore Total expectation complete original mass := sum(1/23, 2/23, 20/23, 0) After Total expectation complete original mass = 1 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Total expectation original atom0 weighted X := mul(1/23, 13) After Total expectation original atom0 weighted X = 13/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Total expectation original atom1 weighted X := mul(2/23, 14) After Total expectation original atom1 weighted X = 28/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Total expectation original atom2 weighted X := mul(20/23, 16) After Total expectation original atom2 weighted X = 320/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Total expectation original atom3 weighted X := mul(0, 112) After Total expectation original atom3 weighted X = 0 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Total expectation full original mean := sum(13/23, 28/23, 320/23, 0) After Total expectation full original mean = 361/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · impliesBefore ORIGINAL masses=[1/23, 2/23, 20/23, 0],SIGNED values X=[13, 14, 16, 112];ORIGINAL cell order=[[0, 1], [2, 3]] After flattened original atom labels=[0, 1, 2, 3] cover EVERY atom exactly ONCE including null atoms;total mass=1;integrable finite mean=361/23;NO independence premise Conditions: none The partition covers all original atoms,including null atoms,exactly once.Keep original cell order and source-normalized internal labels.X may be signed and is integrable in this finite model;independence is not required. Step identifiers and dependenciesRule: totalexp_same_law_complete_disjoint_partition_all_atoms · Uses: G001, S007 S009 · calculationBefore Total expectation cell0 mass := sum(1/23, 2/23) After Total expectation cell0 mass = 3/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Total expectation cell0 numerator := sum(13/23, 28/23) After Total expectation cell0 numerator = 41/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Total expectation cell0 positive-mass conditional mean := div(41/23, 3/23) After Total expectation cell0 positive-mass conditional mean = 41/3 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · impliesBefore ORIGINAL cell index=0,atom labels=[0, 1];mass=3/23;weighted numerator=41/23 After conditional quotient=41/3;positive denominator qualifies division Conditions: none Divide only for positive cell mass.In a null cell,all nonnegative atom masses are0,so the numerator is0 but0/0 remains undefined.Keep null rather than an asserted quotient0,including empty cells. Step identifiers and dependenciesRule: totalexp_cell_quotient_positive_mass_only_null_preserved · Uses: G001, S011 S013 · calculationBefore Total expectation cell1 mass := sum(20/23, 0) After Total expectation cell1 mass = 20/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Total expectation cell1 numerator := sum(320/23, 0) After Total expectation cell1 numerator = 320/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Total expectation cell1 positive-mass conditional mean := div(320/23, 20/23) After Total expectation cell1 positive-mass conditional mean = 16 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · impliesBefore ORIGINAL cell index=1,atom labels=[2, 3];mass=20/23;weighted numerator=320/23 After conditional quotient=16;positive denominator qualifies division Conditions: none Divide only for positive cell mass.In a null cell,all nonnegative atom masses are0,so the numerator is0 but0/0 remains undefined.Keep null rather than an asserted quotient0,including empty cells. Step identifiers and dependenciesRule: totalexp_cell_quotient_positive_mass_only_null_preserved · Uses: G001, S015 S017 · calculationBefore Total expectation positive cell0 weighted mean := mul(3/23, 41/3) After Total expectation positive cell0 weighted mean = 41/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S016 S018 · calculationBefore Total expectation positive cell1 weighted mean := mul(20/23, 16) After Total expectation positive cell1 weighted mean = 320/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · calculationBefore Total expectation sum positive-cell weighted means := sum(41/23, 320/23) After Total expectation sum positive-cell weighted means = 361/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S018 S020 · impliesBefore ALL original cell rows[index,atoms,mass,conditional_or_null,numerator]=[[0, [0, 1], 3/23, 41/3, 41/23], [1, [2, 3], 20/23, 16, 320/23]] After positive-cell weighted sum=361/23;ORIGINAL global mean=361/23;equal=Yes;disjoint complete cells regroup EVERY original p_i*X_i exactly ONCE,null contributions0 Conditions: none Each positive mass times its conditional mean recovers its numerator;null cells contribute0.Disjoint complete cells then regroup every original weighted term exactly once.This proves the finite identity,not an unweighted average or an accidental numerical equality. Step identifiers and dependenciesRule: totalexp_regroup_complete_partition_general_identity · Uses: G001, S019 S021 · calculationBefore Total expectation tower atom0 weighted conditional version := mul(1/23, 41/3) After Total expectation tower atom0 weighted conditional version = 41/69 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S020 S022 · calculationBefore Total expectation tower atom1 weighted conditional version := mul(2/23, 41/3) After Total expectation tower atom1 weighted conditional version = 82/69 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · calculationBefore Total expectation tower atom2 weighted conditional version := mul(20/23, 16) After Total expectation tower atom2 weighted conditional version = 320/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S022 S024 · calculationBefore Total expectation tower atom3 weighted conditional version := mul(0, 16) After Total expectation tower atom3 weighted conditional version = 0 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 S025 · calculationBefore Total expectation full tower expectation := sum(41/69, 82/69, 320/23, 0) After Total expectation full tower expectation = 361/23 Conditions: none Substitute the original roles,order and conditions term by term with exact arithmetic,not floating-point approximation. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S024 S026 · impliesBefore ORIGINAL atom-order conditional version=[41/3, 41/3, 16, 16];ORIGINAL cells=[[0, 1], [2, 3]] After version constant on each cell,so measurable for sigma(partition);each cell integral equals original X cell integral;null-cell version CHOSEN0,not quotient;E(E(X|G))=361/23=E(X)=361/23 Conditions: none Cell-constant values are measurable for the sigma-algebra generated by the partition.Cellwise integral identities extend to its finite cell unions,establishing a conditional-expectation version.On null cells0 is chosen,not a unique quotient.Weight by the original atom masses to obtain the tower identity. Step identifiers and dependenciesRule: totalexp_conditional_version_cell_measurability_and_tower · Uses: G001, S025 ## Reverse checks - S026 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S025 · premise_and_result_audit — Recompute Total expectation full tower expectation with exact rational arithmetic, not rounded decimals. - S024 · premise_and_result_audit — Recompute Total expectation tower atom3 weighted conditional version with exact rational arithmetic, not rounded decimals. - S023 · premise_and_result_audit — Recompute Total expectation tower atom2 weighted conditional version with exact rational arithmetic, not rounded decimals. - S022 · premise_and_result_audit — Recompute Total expectation tower atom1 weighted conditional version with exact rational arithmetic, not rounded decimals. - S021 · premise_and_result_audit — Recompute Total expectation tower atom0 weighted conditional version with exact rational arithmetic, not rounded decimals. - S020 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S019 · premise_and_result_audit — Recompute Total expectation sum positive-cell weighted means with exact rational arithmetic, not rounded decimals. - S018 · premise_and_result_audit — Recompute Total expectation positive cell1 weighted mean with exact rational arithmetic, not rounded decimals. - S017 · premise_and_result_audit — Recompute Total expectation positive cell0 weighted mean with exact rational arithmetic, not rounded decimals. - S016 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S015 · premise_and_result_audit — Recompute Total expectation cell1 positive-mass conditional mean with exact rational arithmetic, not rounded decimals. - S014 · premise_and_result_audit — Recompute Total expectation cell1 numerator with exact rational arithmetic, not rounded decimals. - S013 · premise_and_result_audit — Recompute Total expectation cell1 mass with exact rational arithmetic, not rounded decimals. - S012 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S011 · premise_and_result_audit — Recompute Total expectation cell0 positive-mass conditional mean with exact rational arithmetic, not rounded decimals. - S010 · premise_and_result_audit — Recompute Total expectation cell0 numerator with exact rational arithmetic, not rounded decimals. - S009 · premise_and_result_audit — Recompute Total expectation cell0 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 Total expectation full original mean with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute Total expectation original atom3 weighted X with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute Total expectation original atom2 weighted X with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute Total expectation original atom1 weighted X with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute Total expectation original atom0 weighted X with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute Total expectation complete original mass with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. ## Result ([41/3, 41/3, 16, 16], 361/23, 361/23, Yes, conditional_expectation_constant_on_partition_cells) 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: 2268eaa7e18a32a58ed73937ffa4b4bafd02378b4119ca1d87a57b1136e75eb9 - candidate record SHA-256: 82fa7a2054f3c61a2c9a51345929de55ecdd0b9669790c3d334157202798edf7 - semantic review receipt SHA-256: dd44555041dfbb6fe9936aa00b9a5f8c5982c2784ef61dd0cbd5d1a303a49c1a - question-first witness SHA-256: 9a6011263f6ac53b9a1fd57addf2f3e2f6088b54802c539f332d689480dc2d17 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261109-r04-law-of-total-expectation-0064.json)