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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

$$
\lim _{N \rightarrow \infty} P(a<\text { value }<b)=\int_a^b \frac{1}{\sqrt{2 \pi}} e^{-x^2 / 2} d x
$$

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