Exercise · ZW-CHEBYSHEV-INEQUALITY-009 · ★★★ Advanced

If \(\operatorname{Var}(X)=9\), for which \(\varepsilon\) does the bound \(\sigma^2/\varepsilon^2\) on \(P(|X-\mu|\ge\varepsilon)\) equal exactly \(\tfrac14\)?

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  • B. \(6\)

Explanation

Solve \(9/\varepsilon^2=1/4\): \(\varepsilon^2=36\), \(\varepsilon=6\) (equivalent to \(k=2\) with \(\sigma=3\)).

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Review the concept: Chebyshev inequality

Chebyshev's inequality bounds the probability \(P\) that a variable \(X\) deviates from its mean \(\mu\) by more than \(k\) standard deviations \(\sigma\), using only its variance.

\[P\big(|X-\mu|\ge k\sigma\big)\le\dfrac{1}{k^2}\]

More practice problems: chebyshev inequality

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