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Central Limit Theorem

If i.i.d. x1,xnx_1, \cdots x_n and σx2<\sigma_x^2 < \infty then:

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

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

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

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

V[k=1nXk]=nσx2=n2×σx2n\mathbb{V}[\sum\limits_{k=1}^n X_k] = n \sigma_x^2 = n^2 \times {\sigma_x^2 \over n}