ExampleConstruction: Literature-sourcedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A symmetric two-point distribution attains equality in Chebyshev's inequality

Example

For every $ε > 0$ , let $X$ be uniform on ${- ε, + ε}$ . At the weak threshold $ε$ , equality holds in Chebyshev's inequality.

Facts & Assumptions

Given: A real $ε > 0$ and the random variable in the Example.

[L1]

The two points in a uniform finite space each have probability $1 / 2$ (The uniform probability space on a nonempty finite set).

[L2]

A finite real random variable and its expectation are defined by finite weighted sums (Real random variables on finite probability spaces and their finite distributions, Expectation of a real random variable on a finite probability space).

[L3]

Variance is the expectation of the squared centred variable (Variance, standard deviation, and covariance on a finite probability space).

[L4]

Chebyshev states $P (∣ X - E [X] ∣ \geq t) \leq Var (X) / t^{2}$ for $t > 0$ (Chebyshev's inequality on a finite probability space).

Verification

technique · direct

1.1

Symmetry gives $E [X] = 0$ , and $X^{2} = ε^{2}$ everywhere, so $Var (X) = ε^{2}$ .

L1L2L3algebra

2.1

The event $∣ X - E [X] ∣ \geq ε$ is all of the outcome space and has probability $1$ .

step 1.1L1

3.1

The right side of [L4] at $t = ε$ is $ε^{2} / ε^{2} = 1$ , so equality holds. Positivity of $ε$ licenses the division.

step 1.1step 2.1L4algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 46 results over 16 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

C. M. Grinstead and J. L. Snell, Introduction to Probability, 2nd ed., Section 8.1 after Example 8.1 (standard reference, not scraped)