Exercise · ZW-LAW-OF-LARGE-NUMBERS-002 · ★☆☆ Basic
The variables \(X_i\) are independent with variance \(\sigma^2=25\). What is the variance of \(\bar X_n\) with \(n=100\)?
Choose one option.
Show answer
- C. \(0.25\)
Explanation
For independent variables, \(\operatorname{Var}(\bar X_n)=\sigma^2/n=25/100=0.25\).
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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