Deriving a Markov bound pointwise

Goal and scope

Prove the finite-law bound and compare its exact tail.

A nonnegative random variable with a fully specified finite probability law; not a sampled proof of the general measure-theoretic theorem.

Givens and symbols

• work

• G1 — X takes 0,2,6 with masses 1/2,1/3,1/6; threshold a=3.

• a — positive threshold
• 1_A — event indicator

Rules and sources

• mko_definition · cited — Require X≥0 almost surely, a>0 and finite E[X]; then P(X≥a)≤E[X]/a. Complete finite rational laws allow exact tail and bound calculations. The bound is not the tail probability, signed variables cannot directly use the nonnegative theorem, and threshold0 is excluded. · mko-markov-inequality
• work · prerequisite — Indicator expectations equal event probabilities; nonnegative weighted sums and division by positive numbers preserve order.

Main derivation

1/2+1/3+1/6=1

X\ge0,\quad a=3>0,\quad E[X]<\infty

x<3,\ x\in\{0,2,6\}

3\mathbf1_{\{x\ge3\}}=0\le x

x\ge3

3\mathbf1_{\{x\ge3\}}=3\le x

3\mathbf1_{\{X\ge3\}}\le X

3P(X\ge3)\le E[X]

E[X]=0/2+2/3+6/6

E[X]=5/3

3P(X\ge3)\le5/3

P(X\ge3)\le5/9

\{X\ge3\}=\{X=6\}

P(X\ge3)=1/6<5/9

a\mathbf1_{\{X\ge a\}}\le X,\quad a>0

P(X\ge a)\le E[X]/a

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — This generalization rests on the pointwise proof, not on observed numbers.
• S07 · premise_audit · G1, S06, work — Do not substitute5/9 for the actual event probability.
• S06 · premise_audit · G1, S05, work — The positive threshold makes division valid without reversing order.
• S05 · premise_audit · G1, S04, work — The mass at 6 is 1/6; the three points are not uniform.
• S04 · premise_audit · G1, S03, work — Every support value is used, not a random sample.
• S03 · premise_audit · G1, S02, work — Together the branches exhaust the support.
• S02 · premise_audit · G1, S01, work — A signed x could fail in this branch, locating the nonnegativity requirement.
• S01 · premise_audit · G1, work — A positive mean alone does not replace these qualifications.

Error controls and negative controls

Nonnegativity

Check every positive-mass value, not merely the mean.

Bound versus actual value

Compute tail 1/6 separately from{6}, then compare with 5/9.

• The tail always equals E[X]/a. → reject — Here1/6 is not 5/9.
• Division is valid at a=0. → reject — A positive threshold is a denominator requirement.
• A signed X with positive mean qualifies directly. → reject — The below-threshold branch0≤X can fail.

Result and limitations

The exact tail is 1/6 and the Markov bound5/9; the finite-law version follows pointwise.

Verification and source binding

3 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: c8a078764616e58290cc4d16694339f57787010b9ba42b16d383b7a1a892a42c
• original source SHA-256: 1ef70a0328ec764a48d0df0ab90defc38d6cbe51d3d15f7ef7e50556ddc7cdc8
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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