Exercise · ZW-LAW-OF-LARGE-NUMBERS-009 · ★★★ Advanced
Let \(S_n\) be the number of heads in \(n\) tosses of a fair coin. Which statement does the law of large numbers guarantee?
Choose one option.
Show answer
- B. \(S_n/n\to\tfrac12\)
Explanation
What converges is the proportion \(S_n/n\); the absolute difference \(S_n-n/2\) typically grows like \(\sqrt n\).
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.
\[\bar X_n=\dfrac1n\sum_{i=1}^{n}X_i\ \xrightarrow[n\to\infty]{}\ \mu\]
More practice problems: law of large numbers
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- ZW-LAW-OF-LARGE-NUMBERS-007 With σ^2=9 and n=1000, what bound does Chebyshev give for P(|bar X_n-μ|≥0.3)?
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