Chebyshev Inequality

See how the Chebyshev bound P(|X − 𝔼[X]| ≥ a) ≤ Var(X) / a² compares to the true tail probability.

Distribution

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Deviation a = 25 min from E[X]

520406075

P(|X − E[X]| ≥ a)

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Actual probability

Var(X) / a²

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Chebyshev upper bound

Bound / Actual

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How loose is the bound?

The Cauchy distribution has no finite variance — Chebyshev's inequality does not apply.