Chebyshev inequality

Mathematical statement

Require positive epsilon and finite mean/variance. The event |X−mean|≥epsilon equals (X−mean)^2≥epsilon^2, allowing Markov on the nonnegative squared deviation. The raw bound is not actual tail equality and needs no independence; zero-mass atoms and zero variance retain their almost-sure meaning.

Representative formula (exact TeX)

C(X,e)

Assumptions

• The complete normalized finite rational law and total variable are explicit.
• Epsilon is positive rational, and the mean and second moment are exact.

Symbol scope

• X — finite random variable
• e — positive deviation threshold

Intuition, definition, and misconceptions

Squared deviation is nonnegative.

The Markov threshold is epsilon squared.

Chebyshev requires no independence and supplies only an upper bound.

Do not call the upper bound tail equality or set epsilon=0.

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

• Markov inequality — Apply it to squared centered deviation.
• Variance in finite probability models — The square expectation is the variance.
• Absolute value — The tail uses absolute deviation.

Verification boundary

A complete finite law gives an exact instance, not a finite simulation proving the general theorem.

not_formalized

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

Next obligation

Add total variance and stronger concentration bounds.

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.