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Question 4562: First Mismatch on Positive Probability Support

On this page
  1. Original question and givens
  2. Definition and scope
  3. Step-by-step derivation
  4. Reverse checks
  5. Result
  6. This question review and source binding

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← Question derivation index · Original question batch

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.

Step-by-step derivation

S001 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001

S002 · calculation

Before

probability weight0 := div(0, 14)

After

probability weight0 = 0

Conditions: none

Divide the indexed mass by the common positive denominator D.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S001

S003 · calculation

Before

probability weight1 := div(0, 14)

After

probability weight1 = 0

Conditions: none

Divide the indexed mass by the common positive denominator D.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S002

S004 · calculation

Before

probability weight2 := div(0, 14)

After

probability weight2 = 0

Conditions: none

Divide the indexed mass by the common positive denominator D.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S003

S005 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S004

S006 · calculation

Before

probability weight4 := div(0, 14)

After

probability weight4 = 0

Conditions: none

Divide the indexed mass by the common positive denominator D.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S005

S007 · calculation

Before

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 dependencies

Rule: exact_rational_arithmetic · Uses: G001, S006

S008 · calculation

Before

probability weight6 := div(0, 14)

After

probability weight6 = 0

Conditions: none

Divide the indexed mass by the common positive denominator D.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S007

S009 · calculation

Before

probability weight7 := div(0, 14)

After

probability weight7 = 0

Conditions: none

Divide the indexed mass by the common positive denominator D.

Step identifiers and dependencies

Rule: exact_rational_arithmetic · Uses: G001, S008

S010 · implies

Before

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 dependencies

Rule: probability_variance_scope · Uses: G001, S009

S011 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S010

S012 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S011

S013 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S012

S014 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S013

S015 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S014

S016 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S015

S017 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S016

S018 · implies

Before

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 dependencies

Rule: positive_support_mismatch_row · Uses: G001, S017

S019 · implies

Before

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 dependencies

Rule: 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
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