Chebyshev's Inequality - Sharpness of Bounds

Sharpness of Bounds

As shown in the example above, the theorem will typically provide rather loose bounds. However, the bounds provided by Chebyshev's inequality cannot, in general (remaining sound for variables of arbitrary distribution), be improved upon. For example, for any k ≥ 1, the following example meets the bounds exactly.

X = \begin{cases} -1, & \text{with probability }\frac{1}{2k^2} \\ 0, & \text{with probability }1 - \frac{1}{k^2} \\ 1, & \text{with probability }\frac{1}{2k^2} \end{cases}

For this distribution, mean μ = 0 and standard deviation σ = 1/k, so

\Pr(|X-\mu| \ge k\sigma) = \Pr(|X|\ge1) = \frac{1}{k^2}.

Equality holds only for distributions that are a linear transformation of this one.

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