# Question 20781: The Absolute-Deviation Event Includes Equality at Epsilon [Return to catalogue page 104](https://ocme.evemisslab.com/en/questions/page/104/) [← Question derivation index](https://ocme.evemisslab.com/en/questions/) · [Original question batch](https://ocme.evemisslab.com/en/practice/batch-20261014-advance-20261108-r04-chebyshev-inequality-foundations/#q-20261014-advance-20261108-r04-chebyshev-inequality-0031) 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. R0040-20781 · q-20261014-advance-20261108-r04-chebyshev-inequality-0031 ## Original question and givens Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. 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=3/1. Give (mean, epsilon, tail_probability, certificate). - G001 — Use complete normalized finite law masses=[1/11, 2/11, 8/11] and aligned values X=[0, 6, 9]. 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=3/1. 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](https://ocme.evemisslab.com/en/mko/mko-chebyshev-inequality/) ## Step-by-step derivation S001 · impliesBefore source operation=cheb-tail-probability;original snapshot and mathematics retained After explicit ordered source output fields=[mean, epsilon, tail_probability, certificate] Conditions: none Keep the historical interface marker in the original snapshot while listing source fields here without JavaScript or external lookup.Labels are not mathematical evidence,and No retains its source claim scope. Step identifiers and dependenciesRule: source_ordered_interface_explicit_without_javascript · Uses: G001 S002 · calculationBefore Chebyshev complete probability law mass := sum(1/11, 2/11, 8/11) After Chebyshev complete probability law mass = 1 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S001 S003 · calculationBefore Chebyshev original atom0 weighted SIGNED X := mul(1/11, 0) After Chebyshev original atom0 weighted SIGNED X = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S002 S004 · calculationBefore Chebyshev original atom1 weighted SIGNED X := mul(2/11, 6) After Chebyshev original atom1 weighted SIGNED X = 12/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S003 S005 · calculationBefore Chebyshev original atom2 weighted SIGNED X := mul(8/11, 9) After Chebyshev original atom2 weighted SIGNED X = 72/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S004 S006 · calculationBefore Chebyshev complete SAME-law mean := sum(0, 12/11, 72/11) After Chebyshev complete SAME-law mean = 84/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S005 S007 · calculationBefore Chebyshev original atom0 X minus SAME mean := sub(0, 84/11) After Chebyshev original atom0 X minus SAME mean = -84/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S006 S008 · calculationBefore Chebyshev original atom0 centered SQUARE := mul(-84/11, -84/11) After Chebyshev original atom0 centered SQUARE = 7056/121 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S007 S009 · calculationBefore Chebyshev original atom0 weighted centered square := mul(1/11, 7056/121) After Chebyshev original atom0 weighted centered square = 7056/1331 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S008 S010 · calculationBefore Chebyshev original atom1 X minus SAME mean := sub(6, 84/11) After Chebyshev original atom1 X minus SAME mean = -18/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S009 S011 · calculationBefore Chebyshev original atom1 centered SQUARE := mul(-18/11, -18/11) After Chebyshev original atom1 centered SQUARE = 324/121 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S010 S012 · calculationBefore Chebyshev original atom1 weighted centered square := mul(2/11, 324/121) After Chebyshev original atom1 weighted centered square = 648/1331 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S011 S013 · calculationBefore Chebyshev original atom2 X minus SAME mean := sub(9, 84/11) After Chebyshev original atom2 X minus SAME mean = 15/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S012 S014 · calculationBefore Chebyshev original atom2 centered SQUARE := mul(15/11, 15/11) After Chebyshev original atom2 centered SQUARE = 225/121 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S013 S015 · calculationBefore Chebyshev original atom2 weighted centered square := mul(8/11, 225/121) After Chebyshev original atom2 weighted centered square = 1800/1331 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S014 S016 · calculationBefore Chebyshev full variance E centered square := sum(7056/1331, 648/1331, 1800/1331) After Chebyshev full variance E centered square = 864/121 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S015 S017 · impliesBefore ORIGINAL complete normalized masses=[1/11, 2/11, 8/11];ALL aligned SIGNED values X=[0, 6, 9];SAME-law mean=84/11;all deviations=[-84/11, -18/11, 15/11] After ALL nonnegative squared deviations Y=[7056/121, 324/121, 225/121];variance=E[Y]=864/121;finite moments;variance0 iff X=mean on EVERY POSITIVE-mass atom,not necessarily null atoms Conditions: none This is one random variable under a complete law.X may be negative at positive mass;do not add X≥0 or independence.Each squared Y is nonnegative,and zero variance forces zero deviations only at positive-mass atoms;retain null outliers. Step identifiers and dependenciesRule: cheb_same_law_finite_signed_variable_no_independence_premise · Uses: G001, S016 S018 · calculationBefore Chebyshev original atom0 absolute-event probability mass := mul(1/11, 1) After Chebyshev original atom0 absolute-event probability mass = 1/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S017 S019 · impliesBefore ORIGINAL atom=0;SIGNED centered deviation=-84/11;absolute deviation=84/11;positive epsilon=3 After event abs(deviation)>=epsilon=Yes;EQUALITY INCLUDED;probability contribution=1/11 Conditions: none Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict. Step identifiers and dependenciesRule: cheb_every_absolute_event_atom_including_equality · Uses: G001, S018 S020 · calculationBefore Chebyshev original atom1 absolute-event probability mass := mul(2/11, 0) After Chebyshev original atom1 absolute-event probability mass = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S019 S021 · impliesBefore ORIGINAL atom=1;SIGNED centered deviation=-18/11;absolute deviation=18/11;positive epsilon=3 After event abs(deviation)>=epsilon=No;EQUALITY INCLUDED;probability contribution=0 Conditions: none Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict. Step identifiers and dependenciesRule: cheb_every_absolute_event_atom_including_equality · Uses: G001, S020 S022 · calculationBefore Chebyshev original atom2 absolute-event probability mass := mul(8/11, 0) After Chebyshev original atom2 absolute-event probability mass = 0 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S021 S023 · impliesBefore ORIGINAL atom=2;SIGNED centered deviation=15/11;absolute deviation=15/11;positive epsilon=3 After event abs(deviation)>=epsilon=No;EQUALITY INCLUDED;probability contribution=0 Conditions: none Center at this same law's mean before taking the absolute value.Use ≥ for the event,not >;strict positivity of epsilon does not make the event inequality strict. Step identifiers and dependenciesRule: cheb_every_absolute_event_atom_including_equality · Uses: G001, S022 S024 · calculationBefore Chebyshev COMPLETE absolute-deviation tail probability := sum(1/11, 0, 0) After Chebyshev COMPLETE absolute-deviation tail probability = 1/11 Conditions: none Substitute original roles and established intermediates exactly,retaining indices,set positions,orientation,signs and applicability conditions. Step identifiers and dependenciesRule: exact_rational_arithmetic · Uses: G001, S023 ## 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 (84/11, 3, 1/11, 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. Source hashes and review receipts - input commit: 0f7be61e2c22317a4075561dc822c87d881a50db - original question SHA-256: e087ad252f740934fd9e97bcea0bdfdd6927f5cc159c53af2a7a94173dfd53e5 - candidate record SHA-256: d363916bf85d407228fd63301313258b00c3ce2bdb9bb41e3eb87f577e20ad40 - semantic review receipt SHA-256: c6483ed1229af9183fe5c704a2627b1558ad9a31bfbb0d2ee2d5c1a1c299ab4d - question-first witness SHA-256: e81f9a560d630fcb23441146f008c4feca7aceab3ebe4de2cf1997c061375ef7 - Lean: NOT_RUN · human review: NOT_PERFORMED [Same-source full JSON](https://ocme.evemisslab.com/data/question-derivations/questions/q-20261014-advance-20261108-r04-chebyshev-inequality-0031.json)