Practical Interpretation of Limit Theorems

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Insight:

• Suppose a RV X has mean $\mu$ and standard deviation $\sigma$. Then $S_n=X_1+\cdots +X_n$ has mean $n\mu$ & standard deviation $\sigma \sqrt{n}$ $\overline{X_n}=\frac{X_1+\cdots +X_n}{n}$ has mean $\mu$ & standard deviation $\frac{\sigma }{\sqrt{n}}$

  If X normally distributed, $S_n$ and $\overline{X_n}$ are normally distributed for any $n$ If X not normally distributed, $S_n$ and $\overline{X_n}$ are approximately normally distributed provided that $n$ is sufficiently large

• Note:

  • rule of thumb: $n\ge 30$ to indicate n is large enough. Yet, this depends on the normality of the original value X
  • if n observations chosen without replacement from a finite population, n must also be small compared to population size ⇒ to ensure observations are approximately independent