Illustration of LLN and CLT, the densities of either \(\frac{1}{n}(X_1 + \cdots +X_n -n \mathbb{E}(X_1)) \) or \( \frac{1}{\sqrt{n}}(X_1 + \cdots +X_n-n \mathbb{E}(X_1)) \) are plotted for each choice of i.i.d. distribution \(X_1 \). Recall

1. LLN \(\frac{1}{n}(X_1 + \cdots +X_n ) \to \mathbb{E}(X_1)\).
2. TCL \( \mathbb{P}(\frac{1}{\sqrt{n}\sigma}(X_1 + \cdots +X_n-n \mathbb{E}(X_1)) < a) \to \int_{-\infty}^{a} \frac{1}{\sqrt{2\pi}}e^{-\frac{t^2}{2}}dt \).

Density Simulation