Central Limit Theorem

Let \(\{X_i\}_{i=1}^{N}\) be a sequence of identically and independently distributed random variables with a finite expected value \(\mu\), and a finite and positive variance \(\sigma^2\). For each \(n\), define the random variable \(\overline{X}_n\) as:

\[ \overline{X}_n=\frac{X_1+X_2+\cdots+X_n}{n} \]

Let \(Z\) be a standard normal random variable. For any \(z\in\mathbb{R}\), we have

\[ \lim_{n\to\infty}P\left(\frac{\overline{X}_n-\mu}{\frac{\sigma}{\sqrt{n}}}\le z\right)=P(Z\le z) \]

Informally, CLT states that for sufficiently large \(n\), the random variable

\[ \frac{\overline{X}_n-\mu}{\frac{\sigma}{\sqrt{n}}} \]

is approximately standard normal distributed regardless of the distribution of \(X_i\) and exactly standard normal distributed if \(X_i\) are normally distributed.