Exercise · ZW-LAW-OF-LARGE-NUMBERS-005 · ★☆☆ Basic
What is the expected value of \(\bar X_n\) if each \(X_i\) has expectation \(\mu\)?
Choose one option.
Show answer
- B. \(\mu\)
Explanation
By linearity, \(E[\bar X_n]=\tfrac1n\cdot n\mu=\mu\), regardless of \(n\).
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-008 A fair coin came up heads 5 times in a row. What is the probability of heads on the next toss?
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