TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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Chebyshev's inequality on a finite probability space

Statement

For every finite real random variable $X$ and every $t > 0$ , $P (∣ X - E [X] ∣ \geq t) \leq \frac{Var (X)}{t^{2}} .$ Equivalently, if $σ_{X} > 0$ and $c > 0$ , then $P (∣ X - E [X] ∣ \geq c σ_{X}) \leq 1 / c^{2}$ . If $σ_{X} = 0$ , the first form remains valid for every $t > 0$ .

Facts & Assumptions

Given: A finite real random variable $X$ and a real threshold $t > 0$ .

[L1]

Variance is $E [(X - E [X])^{2}]$ and $σ_{X}^{2} = Var (X)$ (Variance, standard deviation, and covariance on a finite probability space).

[L2]

Markov's inequality states $P (Y \geq a) \leq E [Y] / a$ for nonnegative $Y$ and $a > 0$ (Markov's inequality on a finite probability space).

Proof

technique · direct

1.1givenalgebra

The variable $Y = (X - E [X])^{2}$ is nonnegative, and for $t > 0$ the events ${Y \geq t^{2}}$ and ${∣ X - E [X] ∣ \geq t}$ are equal.

2.1step 1.1L1L2

Apply [L2] to $Y$ at the positive threshold $t^{2}$ and use [L1] to obtain the first inequality.

3.1step 2.1L1choosealgebra∎

If $σ_{X} > 0$ , substitute $t = c σ_{X}$ in step 2.1 and cancel $σ_{X}^{2}$ to obtain $1 / c^{2}$ . Conversely, given $t > 0$ , choose $c = t / σ_{X} > 0$ in the standard-deviation form to recover step 2.1. If $σ_{X} = 0$ , step 2.1 gives probability at most zero for every $t > 0$ .

Depends on

Used by

Dependency tree · two levels

7 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Theorem 8.1 (standard reference, not scraped)
H. Pishro-Nik, Introduction to Probability, Statistics, and Random Processes, Section 6.2.2 (standard reference, not scraped)