Exercise · ZW-CENTRAL-LIMIT-THEOREM-003 · ★☆☆ Basic
36 independent variables, each with mean 2 and variance 9, are added. What is the mean of the sum \(S\)?
Choose one option.
Show answer
- D. \(72\)
Explanation
The expectation of a sum is the sum of the expectations (linearity): \(E[S]=36\cdot2=72\). A typical error is giving the mean of a single term.
Review the concept: Central limit theorem
The central limit theorem states that the average \(\bar X_n\) of \(n\) independent, identically distributed variables \(X\), with mean \(\mu\) and deviation \(\sigma\), once standardized, resembles a normal \(N\).
\[\dfrac{\bar X_n-\mu}{\sigma/\sqrt n}\ \xrightarrow{d}\ N(0,1)\]
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