Why Normal Distribution? Central Limit Theorem

1. Intro

Why is normal distribution so common?

Since the Central Limit Theorem

2. Central Limit Theorem

Random variable $X$

Add N samples of this variable

• The distribution would be a bell curve

How mean and std changes? NX

• $N\times \mu$
• $\sqrt{N}\times \sigma$

No matter what the origin distribution is

• When you normalize it ($\mu =0,\sigma =1$)
• the final result would be a normal distribution

2.1. Why $e$

Since the integral of $e^{-x^2}$ is $\pi$

2.2. Standard

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

2.3. Understanding

$$\underset{N\to \mathrm{\infty }}{lim}P(a<\text{ value }<b)={\int }_a^b\frac{1}{\sqrt{2\pi }}{e}^{-x^2/2}dx$$

Where value is
$$\frac{(X_1+\cdots +X_N)-N\cdot \mu }{\sigma \cdot \sqrt{N}}$$