Exercise · ZW-LAW-OF-LARGE-NUMBERS-003 · ★☆☆ Basic
With \(\sigma^2=1\) and \(n=100\), what bound does Chebyshev give for \(P(|\bar X_n-\mu|\ge0.5)\)?
Choose one option.
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- A. \(0.04\)
Explanation
Apply Chebyshev to \(\bar X_n\), whose variance is \(1/100\): \(\dfrac{1/100}{0.5^2}=0.04\).
Review the concept: Law of large numbers
The law of large numbers states that the average \(\bar X_n\) of \(n\) independent observations of a variable \(X\) approaches its expectation \(\mu\) as \(n\) grows.
\[\bar X_n=\dfrac1n\sum_{i=1}^{n}X_i\ \xrightarrow[n\to\infty]{}\ \mu\]
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