Exercise · ZW-CHEBYSHEV-INEQUALITY-005 · ★★☆ Intermediate
For any variable with finite variance, what minimum probability does Chebyshev guarantee that \(X\) lies within \(2\sigma\) of its mean?
Choose one option.
Show answer
- B. \(\tfrac34\)
Explanation
The complement has probability at most \(1/4\), so the requested one is at least \(1-\tfrac14=\tfrac34\). The 0.95 holds only for the normal.
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
- ZW-CHEBYSHEV-INEQUALITY-001 By Chebyshev, what upper bound holds for P(|X-μ|≥2σ)?
- ZW-CHEBYSHEV-INEQUALITY-002 By Chebyshev, what upper bound holds for P(|X-μ|≥3σ)?
- ZW-CHEBYSHEV-INEQUALITY-003 If X has mean 10 and variance 4, a Chebyshev bound for P(|X-10|≥6) is:
- ZW-CHEBYSHEV-INEQUALITY-004 If Var(X)=tfrac14, the Chebyshev bound for P(|X-μ|≥1) is:
- 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?