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
$$
\lim _{N \rightarrow \infty} P(a<\text { value }<b)=\int_a^b \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} d x
$$
Where value is
$$
\frac{\left(X_1+\cdots+X_N\right)-N \cdot \mu}{\sigma \cdot \sqrt{N}}
$$