Chebyshev from squared deviations

Goal and scope

Derive Chebyshev from Markov and compute an exact instance.

A complete finite law; independence is not required.

Givens and symbols

• work

• G1 — X takes -2,0,2 with masses 1/4,1/2,1/4; epsilon=1.

• μ — mean under this law
• Y — squared deviation
• ε — positive threshold

Rules and sources

• mko_definition · cited — 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. · mko-chebyshev-inequality
• work · prerequisite — Assumed prerequisite Markov: for nonnegative Y and a>0, P(Y≥a)≤E[Y]/a; variance is the centered second moment.

Main derivation

1/4+1/2+1/4=1

\mu=E[X]=-2/4+2/4=0

Y=(X-\mu)^2

Y\in\{4,0,4\},\quad Y\ge0

E[Y]=4/4+0/2+4/4

\operatorname{Var}(X)=E[Y]=2

\varepsilon>0

|X-\mu|\ge\varepsilon\iff Y\ge\varepsilon^2

P(Y\ge\varepsilon^2)\le E[Y]/\varepsilon^2

P(|X-\mu|\ge\varepsilon)\le\operatorname{Var}(X)/\varepsilon^2

\varepsilon=1

\operatorname{Var}(X)/\varepsilon^2=2

\{|X|\ge1\}=\{-2,2\}

P(|X|\ge1)=1/2\le2

\varepsilon=2

P(|X|\ge2)=1/2=2/2^2

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · G1, S07, work — This is equality for this law, not a universal equality claim.
• S07 · premise_audit · G1, S06, work — Zero is excluded; the event retains≥.
• S06 · premise_audit · G1, S05, work — One may clip to min(1,2), but cannot claim a tail of 2.
• S05 · premise_audit · G1, S04, work — Y nonnegative and finite expectation have already been established.
• S04 · premise_audit · G1, S03, work — Positive epsilon is necessary before division by its square.
• S03 · premise_audit · G1, S02, work — Center at mu=0, not an observation or another law's mean.
• S02 · premise_audit · G1, S01, work — Y retains the masses from the original X law.
• S01 · premise_audit · G1, work — Signed X is allowed; Markov will apply to its square.

Error controls and negative controls

Squared event

Establish the event equivalence before invoking Markov.

Same-law center

Use the same complete law for the mean, Y and variance.

• Chebyshev needs independent variables. → reject — The proof uses only one X and its second moment.
• The tail is 2 here. → reject — Two is a loose bound; the tail is 1/2.
• The formula applies at epsilon=0. → reject — Its denominator epsilon squared cannot be zero.

Result and limitations

Chebyshev follows by Markov on squared deviations; at epsilon=1 the exact tail is 1/2.

Verification and source binding

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

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: 0bd2da48a49fb5d3544edcc75194ecfe281101c23dfc66699760f407b58858d2
• original source SHA-256: 740b165db0bc59f0b2d318f01da05196eceb990ec8921f46642cf8407f1c0144
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

Same-source full JSON (formulas, ASTs, controls and revision log)