Central Limit Theorem

If i.i.d. $x_1, \cdots x_n$ and $\sigma_x^2 < \infty$ then:

$\text{std}(\overline{x_n}) \rightarrow^d \mathbb{Z} \sim \mathbb{N}(0,~1)$

$\text{std}(\sum\limits_{k=1}^n{x_n}) \rightarrow^d \mathbb{Z} \sim \mathbb{N}(0,~1)$

$\text{std}(\overline{x_n}) \rightarrow {{\overline{X_n} - \mu_x} \over {\sigma_x \over \sqrt{n}}}$

${\sum\limits_{k=1}^n X_n - n\mu_x} \over \sigma_x \sqrt{n}$

$\mathbb{V}[\sum\limits_{k=1}^n X_k] = n \sigma_x^2 = n^2 \times {\sigma_x^2 \over n}$