Exercise · ZW-CHEBYSHEV-INEQUALITY-003 · ★☆☆ Basic
If \(X\) has mean 10 and variance 4, a Chebyshev bound for \(P(|X-10|\ge6)\) is:
Choose one option.
Show answer
- C. \(\tfrac19\)
Explanation
With \(\sigma=2\), \(6=3\sigma\), so \(k=3\) and the bound is \(1/9\). Equivalently \(\sigma^2/6^2=4/36\).
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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- ZW-CHEBYSHEV-INEQUALITY-004 If Var(X)=tfrac14, the Chebyshev bound for P(|X-μ|≥1) is:
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- ZW-CHEBYSHEV-INEQUALITY-006 If X has mean 100 and standard deviation 10, what minimum probability does Chebyshev guarantee for…
- ZW-CHEBYSHEV-INEQUALITY-007 What must be known about X to apply the Chebyshev inequality?
- ZW-CHEBYSHEV-INEQUALITY-008 What is the smallest k for which Chebyshev guarantees P(|X-μ|≥ kσ)≤0.04?
- ZW-CHEBYSHEV-INEQUALITY-009 If Var(X)=9, for which ε does the bound σ^2/ε^2 on P(|X-μ|≥ε) equal exactly tfrac14?