Question 20211: A Markov Upper Bound Is Not the Tail Probability

Original question and givens

Use complete normalized finite law with exact masses=[1/16, 2/16, 13/16] and aligned total values X=[0, 11, 13]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=12>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, certificate).

• G001 — Use complete normalized finite law with exact masses=[1/16, 2/16, 13/16] and aligned total values X=[0, 11, 13]. Each mass is nonnegative and the masses sum to1. All moments here are finite. The prototype checks every value for its pointwise nonnegativity flag; a null-mass negative value is not by itself failure of the theorem's almost-sure premise. For this theorem exercise all table values are nonnegative and threshold=12>0. Retain the raw upper bound without clipping it to1. Give (tail_probability, expectation, raw_upper_bound, bound_holds, 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-markov-inequality

Step-by-step derivation

source operation=markov-bound-comparison;original snapshot and mathematics retained

explicit ordered source output fields=[tail_probability, expectation, raw_upper_bound, bound_holds, certificate]

Markov complete probability mass := sum(1/16, 1/8, 13/16)

Markov complete probability mass = 1

Markov original atom0 weighted X := mul(1/16, 0)

Markov original atom0 weighted X = 0

Markov original atom1 weighted X := mul(1/8, 11)

Markov original atom1 weighted X = 11/8

Markov original atom2 weighted X := mul(13/16, 13)

Markov original atom2 weighted X = 169/16

Markov complete SAME-law expectation := sum(0, 11/8, 169/16)

Markov complete SAME-law expectation = 191/16

ORIGINAL complete nonempty masses=[1/16, 1/8, 13/16];ALL aligned values X=[0, 11, 13];full mass=1;ALL weighted X terms=[0, 11/8, 169/16]

finite exact expectation=191/16;prototype EVERY-ENTRY nonnegative=Yes;actual positive-mass nonnegativity=Yes;null-negative atoms=[];positive-mass negative atoms=[]

Markov original atom0 threshold-event mass := mul(1/16, 0)

Markov original atom0 threshold-event mass = 0

ORIGINAL atom=0;value X=0;positive threshold=12;original mass=1/16

event X>=threshold=No;EQUALITY INCLUDED;event mass contribution=0

Markov original atom1 threshold-event mass := mul(1/8, 0)

Markov original atom1 threshold-event mass = 0

ORIGINAL atom=1;value X=11;positive threshold=12;original mass=1/8

event X>=threshold=No;EQUALITY INCLUDED;event mass contribution=0

Markov original atom2 threshold-event mass := mul(13/16, 1)

Markov original atom2 threshold-event mass = 13/16

ORIGINAL atom=2;value X=13;positive threshold=12;original mass=13/16

event X>=threshold=Yes;EQUALITY INCLUDED;event mass contribution=13/16

Markov complete exact tail probability := sum(0, 0, 13/16)

Markov complete exact tail probability = 13/16

ALL original threshold-event flags=[No, No, Yes];ALL law-weighted indicator terms=[0, 0, 13/16]

exact complete-law tail probability=13/16;this event calculation ALONE does not assert Markov applicability

Markov RAW expectation over positive threshold := div(191/16, 12)

Markov RAW expectation over positive threshold = 191/192

Markov proof atom0 threshold times indicator := mul(12, 0)

Markov proof atom0 threshold times indicator = 0

Markov proof atom0 pointwise X minus threshold indicator := sub(0, 0)

Markov proof atom0 pointwise X minus threshold indicator = 0

Markov proof atom0 law-weighted nonnegative slack := mul(1/16, 0)

Markov proof atom0 law-weighted nonnegative slack = 0

Markov proof atom1 threshold times indicator := mul(12, 0)

Markov proof atom1 threshold times indicator = 0

Markov proof atom1 pointwise X minus threshold indicator := sub(11, 0)

Markov proof atom1 pointwise X minus threshold indicator = 11

Markov proof atom1 law-weighted nonnegative slack := mul(1/8, 11)

Markov proof atom1 law-weighted nonnegative slack = 11/8

Markov proof atom2 threshold times indicator := mul(12, 1)

Markov proof atom2 threshold times indicator = 12

Markov proof atom2 pointwise X minus threshold indicator := sub(13, 12)

Markov proof atom2 pointwise X minus threshold indicator = 1

Markov proof atom2 law-weighted nonnegative slack := mul(13/16, 1)

Markov proof atom2 law-weighted nonnegative slack = 13/16

Markov complete nonnegative weighted slack sum := sum(0, 11/8, 13/16)

Markov complete nonnegative weighted slack sum = 35/16

Markov threshold times exact tail := mul(12, 13/16)

Markov threshold times exact tail = 39/4

Markov expectation minus threshold tail := sub(191/16, 39/4)

Markov expectation minus threshold tail = 35/16

ALL qualified original values nonnegative;for each atom threshold*1_event<=X;weighted slack terms=[0, 11/8, 13/16];threshold=12>0

E[X]-threshold*P(event)=35/16>=0;divide POSITIVE threshold to obtain actual tail=13/16<=RAW bound=191/192;NO equality claim and NO clipping to1

Reverse checks

• S029 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S028 · premise_and_result_audit — Recompute Markov expectation minus threshold tail with exact rational arithmetic, not rounded decimals.
• S027 · premise_and_result_audit — Recompute Markov threshold times exact tail with exact rational arithmetic, not rounded decimals.
• S026 · premise_and_result_audit — Recompute Markov complete nonnegative weighted slack sum with exact rational arithmetic, not rounded decimals.
• S025 · premise_and_result_audit — Recompute Markov proof atom2 law-weighted nonnegative slack with exact rational arithmetic, not rounded decimals.
• S024 · premise_and_result_audit — Recompute Markov proof atom2 pointwise X minus threshold indicator with exact rational arithmetic, not rounded decimals.
• S023 · premise_and_result_audit — Recompute Markov proof atom2 threshold times indicator with exact rational arithmetic, not rounded decimals.
• S022 · premise_and_result_audit — Recompute Markov proof atom1 law-weighted nonnegative slack with exact rational arithmetic, not rounded decimals.
• S021 · premise_and_result_audit — Recompute Markov proof atom1 pointwise X minus threshold indicator with exact rational arithmetic, not rounded decimals.
• S020 · premise_and_result_audit — Recompute Markov proof atom1 threshold times indicator with exact rational arithmetic, not rounded decimals.
• S019 · premise_and_result_audit — Recompute Markov proof atom0 law-weighted nonnegative slack with exact rational arithmetic, not rounded decimals.
• S018 · premise_and_result_audit — Recompute Markov proof atom0 pointwise X minus threshold indicator with exact rational arithmetic, not rounded decimals.
• S017 · premise_and_result_audit — Recompute Markov proof atom0 threshold times indicator with exact rational arithmetic, not rounded decimals.
• S016 · premise_and_result_audit — Recompute Markov RAW expectation over positive threshold with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Markov complete exact tail probability with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Markov original atom2 threshold-event mass with exact rational arithmetic, not rounded decimals.
• 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 — Recompute Markov original atom1 threshold-event mass with exact rational arithmetic, not rounded decimals.
• S009 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S008 · premise_and_result_audit — Recompute Markov original atom0 threshold-event mass with exact rational arithmetic, not rounded decimals.
• S007 · premise_and_result_audit — Check against the original data and this step's premises; a one-way implication is not an equivalence.
• S006 · premise_and_result_audit — Recompute Markov complete SAME-law expectation with exact rational arithmetic, not rounded decimals.
• S005 · premise_and_result_audit — Recompute Markov original atom2 weighted X with exact rational arithmetic, not rounded decimals.
• S004 · premise_and_result_audit — Recompute Markov original atom1 weighted X with exact rational arithmetic, not rounded decimals.
• S003 · premise_and_result_audit — Recompute Markov original atom0 weighted X with exact rational arithmetic, not rounded decimals.
• S002 · premise_and_result_audit — Recompute Markov complete probability 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

(13/16, 191/16, 191/192, Yes, upper_bound_not_tail_equality)

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.