Question 20798: The Absolute-Deviation Event Includes Equality at Epsilon

Original question and givens

Use complete normalized finite law masses=[1/28, 2/28, 25/28] and aligned values X=[0, 23, 26]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, certificate).

• G001 — Use complete normalized finite law masses=[1/28, 2/28, 25/28] and aligned values X=[0, 23, 26]. Masses are nonnegative and sum to1; values may be signed. Mean and variance use the same law. No independence premise is required for this one-variable Chebyshev bound; zero-mass values remain in the table but contribute no event mass or moment. Positive epsilon=2/1. Give (mean, epsilon, tail_probability, 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-chebyshev-inequality

Step-by-step derivation

source operation=cheb-tail-probability;original snapshot and mathematics retained

explicit ordered source output fields=[mean, epsilon, tail_probability, certificate]

Chebyshev complete probability law mass := sum(1/28, 1/14, 25/28)

Chebyshev complete probability law mass = 1

Chebyshev original atom0 weighted SIGNED X := mul(1/28, 0)

Chebyshev original atom0 weighted SIGNED X = 0

Chebyshev original atom1 weighted SIGNED X := mul(1/14, 23)

Chebyshev original atom1 weighted SIGNED X = 23/14

Chebyshev original atom2 weighted SIGNED X := mul(25/28, 26)

Chebyshev original atom2 weighted SIGNED X = 325/14

Chebyshev complete SAME-law mean := sum(0, 23/14, 325/14)

Chebyshev complete SAME-law mean = 174/7

Chebyshev original atom0 X minus SAME mean := sub(0, 174/7)

Chebyshev original atom0 X minus SAME mean = -174/7

Chebyshev original atom0 centered SQUARE := mul(-174/7, -174/7)

Chebyshev original atom0 centered SQUARE = 30276/49

Chebyshev original atom0 weighted centered square := mul(1/28, 30276/49)

Chebyshev original atom0 weighted centered square = 7569/343

Chebyshev original atom1 X minus SAME mean := sub(23, 174/7)

Chebyshev original atom1 X minus SAME mean = -13/7

Chebyshev original atom1 centered SQUARE := mul(-13/7, -13/7)

Chebyshev original atom1 centered SQUARE = 169/49

Chebyshev original atom1 weighted centered square := mul(1/14, 169/49)

Chebyshev original atom1 weighted centered square = 169/686

Chebyshev original atom2 X minus SAME mean := sub(26, 174/7)

Chebyshev original atom2 X minus SAME mean = 8/7

Chebyshev original atom2 centered SQUARE := mul(8/7, 8/7)

Chebyshev original atom2 centered SQUARE = 64/49

Chebyshev original atom2 weighted centered square := mul(25/28, 64/49)

Chebyshev original atom2 weighted centered square = 400/343

Chebyshev full variance E centered square := sum(7569/343, 169/686, 400/343)

Chebyshev full variance E centered square = 2301/98

ORIGINAL complete normalized masses=[1/28, 1/14, 25/28];ALL aligned SIGNED values X=[0, 23, 26];SAME-law mean=174/7;all deviations=[-174/7, -13/7, 8/7]

ALL nonnegative squared deviations Y=[30276/49, 169/49, 64/49];variance=E[Y]=2301/98;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms

Chebyshev original atom0 absolute-event probability mass := mul(1/28, 1)

Chebyshev original atom0 absolute-event probability mass = 1/28

ORIGINAL atom=0;SIGNED centered deviation=-174/7;absolute deviation=174/7;positive epsilon=2

event abs(deviation)>=epsilon=Yes;EQUALITY INCLUDED;probability contribution=1/28

Chebyshev original atom1 absolute-event probability mass := mul(1/14, 0)

Chebyshev original atom1 absolute-event probability mass = 0

ORIGINAL atom=1;SIGNED centered deviation=-13/7;absolute deviation=13/7;positive epsilon=2

event abs(deviation)>=epsilon=No;EQUALITY INCLUDED;probability contribution=0

Chebyshev original atom2 absolute-event probability mass := mul(25/28, 0)

Chebyshev original atom2 absolute-event probability mass = 0

ORIGINAL atom=2;SIGNED centered deviation=8/7;absolute deviation=8/7;positive epsilon=2

event abs(deviation)>=epsilon=No;EQUALITY INCLUDED;probability contribution=0

Chebyshev COMPLETE absolute-deviation tail probability := sum(1/28, 0, 0)

Chebyshev COMPLETE absolute-deviation tail probability = 1/28

Reverse checks

• S024 · premise_and_result_audit — Recompute Chebyshev COMPLETE absolute-deviation tail probability with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S022 · premise_and_result_audit — Recompute Chebyshev original atom2 absolute-event probability mass with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S020 · premise_and_result_audit — Recompute Chebyshev original atom1 absolute-event probability mass with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Chebyshev original atom0 absolute-event probability mass with exact rational arithmetic, not rounded decimals.
• 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 Chebyshev full variance E centered square with exact rational arithmetic, not rounded decimals.
• S015 · premise_and_result_audit — Recompute Chebyshev original atom2 weighted centered square with exact rational arithmetic, not rounded decimals.
• S014 · premise_and_result_audit — Recompute Chebyshev original atom2 centered SQUARE with exact rational arithmetic, not rounded decimals.
• S013 · premise_and_result_audit — Recompute Chebyshev original atom2 X minus SAME mean with exact rational arithmetic, not rounded decimals.
• S012 · premise_and_result_audit — Recompute Chebyshev original atom1 weighted centered square with exact rational arithmetic, not rounded decimals.
• S011 · premise_and_result_audit — Recompute Chebyshev original atom1 centered SQUARE with exact rational arithmetic, not rounded decimals.
• S010 · premise_and_result_audit — Recompute Chebyshev original atom1 X minus SAME mean with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Recompute Chebyshev original atom0 weighted centered square with exact rational arithmetic, not rounded decimals.
• S008 · premise_and_result_audit — Recompute Chebyshev original atom0 centered SQUARE with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Recompute Chebyshev original atom0 X minus SAME mean with exact rational arithmetic, not rounded decimals.
• S006 · premise_and_result_audit — Recompute Chebyshev complete SAME-law mean with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Chebyshev original atom2 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Chebyshev original atom1 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Chebyshev original atom0 weighted SIGNED X with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Chebyshev complete probability law 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

(174/7, 2, 1/28, exact_absolute_deviation_tail)

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.