Central Limit Theorem

$\sqrt{N}{\hat{X}}_N\to \mathcal{N}(\mu ,{\sigma }^2)$ as $N\to \infty$, where

• $X_i$ are i.i.d. Random variables
• ${\hat{X}}_N:=\frac{1}{N}{\sum }_i^N{X}_i$
• $\mu :=\mathbb{E}[X_i]$
• $\sigma :=\mathrm{Var}(X_i)$