Central Limit Theorem

When random effects of many independant small sized causes sum up to large scale observable effects : one gets the Normal Distribution
Let $(X_{i})_{i \in N}$ is a seq of independant, real valued, [[X_{i}- E[X_{i}|(X_{i}- E[X_{i}]] = E[[X_{i}- E[X_{i}|[X_{i}- E[X_{i}]] $P_{S_{n}}$ of standardized sum variables converge weakly to $N (0, 1∣ [Sq u a re I n t e g r ab l e]$ . $S_{n} = \frac{Σ _{i = 1}^{n} ( X _{i} - E [ X _{i} ])}{σ ( Σ _{i = 1}^{n} X _{i} )}$
- Converge weakly : $l i m_{n \to \infty} \int f (x) P_{n} (d x) = \int f (x) P (d x)$ for all $f : R \to R$
- Lebesgue Integrals

$X_{i}$ Are Identically Distributed

Regardless of shape of each $X_{i}$ , distribution of normalized sum converges to $N (0, 1)$
Uniformly bounded
None of the $X_{i}$ dominates the other “washing out”