Markov inequality

Mathematical statement

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.

Representative formula (exact TeX)

M(X,a)

Assumptions

• The complete normalized finite rational law and total variable-value table are supplied.
• Main theorem exercises require every table value nonnegative and the threshold positive.

Symbol scope

• X — nonnegative random variable
• a — positive threshold

Intuition, definition, and misconceptions

Nonnegativity, a positive threshold and finite expectation are explicit hypotheses.

The actual tail is a sum over the complete law.

Markov supplies an upper bound, not equality.

Do not apply the nonnegative theorem to a signed variable or call E[X]/a the actual probability.

Adopted derivation guide

The original MKO is retained. The guide has its own append-only revision and verification boundaries, not a new Lean proof.

Prerequisites

• Real random variables on finite sample spaces — X and its law are specified.
• Expectation in finite models — The bound uses E[X].
• Probability on finite sample spaces — The event is a threshold tail.
• Usual order on the real numbers — Nonnegativity and the threshold comparison use the order.

Verification boundary

Exact exercises use complete finite laws; the general measure version is stated, not proved by finite samples. The prototype pointwise check is stronger than almost-sure nonnegativity at null-mass entries; ambiguous negative-null cases need source review.

not_formalized

This does not mean the content is false; it means no replayable computational or formal evidence is registered yet.

Next obligation

Add Chebyshev inequality.

Same-source structured data

This page and the interactive view use the same object and existing language edition; this adds no proof or human review.