# Question 4562: First Mismatch on Positive Probability Support [Return to catalogue page 23](https://ocme.evemisslab.com/en/questions/page/23/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261008-advance-20261010-r06-variance-foundations/#q-20261008-advance-20261010-r06-variance-0087) 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. R0008-04562 · q-20261008-advance-20261010-r06-variance-0087 ## Original question and givens S={0,…,7}, F=P(S), masses=[0, 0, 0, 12, 0, 2, 0, 0], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 12, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0. - G001 — S={0,…,7}, F=P(S), masses=[0, 0, 0, 12, 0, 2, 0, 0], D=14, and w_i=mass_i/D. Masses are nonnegative and sum exactly to D; uniform weights are not assumed. X=[100, 101, 102, 0, 104, 12, 106, 107] in index order. Use the X value at the first positive-mass index as reference. Return the first positive-mass index with a different X value, or none; indices start at 0. ## 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-variance](https://ocme.evemisslab.com/en/mko/mko-variance/) ## Step-by-step derivation S001 · calculationBefore total mass numerator := sum(0, 0, 0, 12, 0, 2, 0, 0) After total mass numerator = 14 Conditions: none Nonnegative masses must sum to D to give total probability1. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001 S002 · calculationBefore probability weight0 := div(0, 14) After probability weight0 = 0 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore probability weight1 := div(0, 14) After probability weight1 = 0 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore probability weight2 := div(0, 14) After probability weight2 = 0 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore probability weight3 := div(12, 14) After probability weight3 = 6/7 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore probability weight4 := div(0, 14) After probability weight4 = 0 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore probability weight5 := div(2, 14) After probability weight5 = 1/7 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore probability weight6 := div(0, 14) After probability weight6 = 0 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore probability weight7 := div(0, 14) After probability weight7 = 0 Conditions: none Divide the indexed mass by the common positive denominator D. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · impliesBefore values=[100, 101, 102, 0, 104, 12, 106, 107];weights=[0/1, 0/1, 0/1, 6/7, 0/1, 1/7, 0/1, 0/1] After known finite probability law,not a sampled estimate Conditions: none This is the complete normalized probability law,so variance uses its probability weights,not an n−1 sample estimate.Retain original indices of zero-mass outcomes. Step identifiers and dependenciesRule: probability_variance_scope · Uses: G001, S009 S011 · impliesBefore index=0;mass=0;X=100;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S010 S012 · impliesBefore index=1;mass=0;X=101;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S011 S013 · impliesBefore index=2;mass=0;X=102;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S012 S014 · impliesBefore index=3;mass=12;X=0;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S013 S015 · impliesBefore index=4;mass=0;X=104;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S014 S016 · impliesBefore index=5;mass=2;X=12;anchor X=0 After positive-mass mismatch=Yes Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S015 S017 · impliesBefore index=6;mass=0;X=106;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S016 S018 · impliesBefore index=7;mass=0;X=107;anchor X=0 After positive-mass mismatch=No Conditions: none A mismatch requires both positive mass and a different value.Arbitrary values at zero-weight outcomes do not change this positive-support test. Step identifiers and dependenciesRule: positive_support_mismatch_row · Uses: G001, S017 S019 · impliesBefore first positive index=3;all eligible mismatch indices=[5] After 5 Conditions: none Choose the first original0-based mismatch index,not the smallest X-value.No mismatch means X is constant on its positive support. Step identifiers and dependenciesRule: first_positive_support_mismatch · Uses: G001, S018 ## 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S013 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - 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 — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S010 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence. - S009 · premise_and_result_audit — Recompute probability weight7 with exact rational arithmetic, not rounded decimals. - S008 · premise_and_result_audit — Recompute probability weight6 with exact rational arithmetic, not rounded decimals. - S007 · premise_and_result_audit — Recompute probability weight5 with exact rational arithmetic, not rounded decimals. - S006 · premise_and_result_audit — Recompute probability weight4 with exact rational arithmetic, not rounded decimals. - S005 · premise_and_result_audit — Recompute probability weight3 with exact rational arithmetic, not rounded decimals. - S004 · premise_and_result_audit — Recompute probability weight2 with exact rational arithmetic, not rounded decimals. - S003 · premise_and_result_audit — Recompute probability weight1 with exact rational arithmetic, not rounded decimals. - S002 · premise_and_result_audit — Recompute probability weight0 with exact rational arithmetic, not rounded decimals. - S001 · premise_and_result_audit — Recompute total mass numerator with exact rational arithmetic, not rounded decimals. ## Result 5 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: 5d0440e37a95997de7a54ab50eee519c23c86cc32d49b4ca2cb032ce786c6e1d - candidate record SHA-256: 3ee57631bc68b4de71e548ac0591994908817d27dbe8465ca3a2067053aed0ac - semantic review receipt SHA-256: b6e05a3e8193a93db505dbf337528fca72e2e41528b1e92b698a3f0683d74289 - question-first witness SHA-256: caa8693baf0b91acd09a20215d1aa6954ae513fc0b0814f26ac1c8c834b60d67 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261008-advance-20261010-r06-variance-0087.json)