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2.2.2 The Central Limit Theorem (CLT)

Theorem 1   If $ X_1,X_2,\ldots$ are i.i.d. with common mean $ \mu$ and common variance $ \sigma^2$, then:

$\displaystyle \frac{\sum_{k=1}^{n} (X_k - \mu)}{\sigma \sqrt{n}}$

has distribution converging to $ N(0,1)$ (``the standard normal'').

Hence:

\begin{equation*}i \begin{split}P(a \le \frac{\sum_{k=1}^{n} (X_k - \mu)}{\sigma...
...\sqrt{2\pi}} e^{-\frac{x^2}{2}} dx \\ P(a\le Z \le b) \end{split}\end{equation*} (2)

where $ Z \sim N(0,1)$. Remarks:
  1. If $ Z \sim N(0,1)$ then $ aZ + b \sim N(b,a^2)$.
  2. For large $ n$, $ \frac{\sum_{k=1}^{n} (X_k - \mu)}{\sigma \sqrt{n}}\approx N(0,1)$.
  3. For large $ n$, $ \sum_{k=1}^n X_k \approx N(n\mu, n\sigma^2)$.



Eran Borenstein
2007-05-04