The Normal Distribution

Weak law of large numbers

Let $X_1,\dots ,X_n$

Strong Law of large numbers

$$f(X)=\frac{1}{\sigma \sqrt{2\pi }}{e}^{-\frac{1}{2}{\left(\frac{X-\mu }{\sigma }\right)}^2}$$

• Standard Normal Distribution

$$f(X)=\frac{1}{\sqrt{2\pi }}{e}^{-\frac{1}{2}{X}^2}$$ $$X\sim N(0,1)$$

• Distribution of $\overline{X}$ for a Normal Law

  • If $X_1\dots X_n$ follow normal distribution then the their mean $\overline{X}$ also follows a normal distribution $N({\mu }_X,\frac{{\sigma }_X^2}{2})$

• Central limit theorem: Let $X_1\dots X_n$ follow any distribution (Chi-sq, exponential, gamma etc). The mean of the samples is going to follow a normal distribution.

• A regression is a implementation of a “conditional mean”.

• The central limit theorem is really about sample averages and not about individual observations.