# Theorem Central Limit - An Illustration

The central limit theorem (CLT) states that the average of independent random variables tends toward a normal distribution even if the original variables themselves are not normally distributed.

Theorem

Let $$\{X_1 \dots X_n\}$$ be a sequence of $$n$$ independent and identically distributed random variables drawn from distributions of expected values given by $$\mu$$ and finite variances given by $$\sigma^2$$. Let $$S_n$$ the sample average: $S_n = \frac{X_1 + X_2 + \dots + X_n}{n}$ Then as $$n$$ tends to the infinite, we have the convergence (in distribution) of $$S_n$$ towards a normal distribution: $\frac{S_n-\mu}{\frac{\sigma}{\sqrt{n}}} \longrightarrow \mathcal{N}(0,1)$

We are running 1000 times the sum of n draws in an uniform distribution $$\mathcal{U}(0,1)^{*}$$.
Select n :

#### Simulated distributions

$$^{*}$$ Mean of $$\;\mathcal{U}(0,1)$$ is $$\mu = 0.5$$, standard deviation is $$\sigma = \frac{1}{2\sqrt{3}}$$.