Central Limit Theorem

Definition

The Standardized Average of Independent Random Variable with finite means and variance converge to Normal Distribution $\text{Normal}(0,1)$

Formal Definition

$Z_n=\sqrt{n}\left(\frac{{\overline{X}}_n-\mu }{\sigma }\right){\to }_D\text{Normal}(0,1)⟺{\overline{X}}_n{\sim }_{appr}N(\mu ,\frac{{\sigma }^2}{n})$

Normal Curve Definition

${\lim }_{N\to \infty }P\left(a<\frac{{\overline{X}}_n-N\ast \mu }{\sigma \ast \sqrt{N}}<b\right)={\int }_b^a\frac{1}{\sqrt{2\pi }}{e}^{-x^2/\ast 2\ast }dx$

Central Limit Theorem Approximation Examples

Example 1

Example 2