# Central Limit Theorem CLT states that the distribution of $\bar X_n$ after standardization approaches a ==standard Normal distribution==. > [!tip] Central Limit Theorem > > As $n\to\infty$, > > $ > \sqrt n \left(\frac{\bar X_n - \mu}\sigma\right) \to \mathcal N(0, 1) > $ > > in distribution. Regardless the distribution of $X_j$, _averaging_ these r.v.s will always lead to _normality_. Consider the sum of the i.i.d. r.v.s, we also have $ W_n \sim \mathcal N(n\mu, n\sigma^2) $ Which is useful for approximating sum of i.i.d. r.v.s. - Poisson: $Y \sim \mathcal N(n, n)$ - Binomial: $Y \sim \mathcal N(np, np(1 - p))$ Cauchy distribution has no true mean and variance, and CLT and [[lln|LLN]] doesn't hold.