normal distribution

All you need to know about Gaussian distribution

The distribution is fully described by the paramteres $\mu$ and $\sigma$, usually denoted as $\mathcal{N}(\mu ,{\sigma }^2)$, where $\mu$ represents the expected value of the distribution and $\sigma$ is the standard deviation.
The density is symmetrical across the expected value.
Main characteristic: Fluctuations around a central value.

For a higher-level description, see also: multivariate gaussian distribution.

Visualize: Open desmos and type normaldist(mu, var), to visualize a normal distribution with mean, variance, and cumulative probability.

Intuition

$e^x$ is exponential growth.
$e^{-x}$ is exponential decay.
$e^{-|x|}$ is exponential growth until $x=0$, where there is a sharp point and it turns to exp. decay
$e^{-x^2}$ smooths the entire function, which still decays in both directions (seen from the top), a bell curve.
$e^{-c{x}^2}$ the constant adjusts for wider / narrower bell curves. Can be rearranged to $(e^c{)}^{x^2}$ → $e$ not special.
Dividing $x$ by sigma and after squaring by $2$ allows us to specify the standard deviation.
Subtracting $\mu$ let’s us shift the distribution left and right.
After dividing by $\sqrt{\pi }$, the area under the curve equals $1$.
We need to divide by one half sigma aswell to keep the area. Final result:

$$\text{PDF: }f(x)=\underset{\text{Area = 1}}{\underbrace{\frac{1}{\sigma \sqrt{2\pi }}}}\exp (\frac{1}{2}{\left(\frac{x-\mu }{\sigma }\right)}^2)$$

If $\sigma =1$ it is called the standard normal distribution.

Visually: 3b1b

Arises when many different independent variables are summed.

Affine property

Any affine transformation $f(x)=Ax+b$ of a gaussian $X\sim \mathcal{N}(\mu ,{\sigma }^2)$ is also a gaussian.

$$Y=aX+b\sim \mathcal{N}(a\mu +b,a^2{\sigma }^2)$$

So for example:
$\mathcal{N}(m,C)\sim m+\mathcal{N}(0,C)$
$\sigma N(m,C)\sim \mathcal{N}(m,{\sigma }^{2}C)$
$AN(m,I)\sim N(m,A{A}^T)$

Code

“From scratch”

import numpy as np
import matplotlib.pyplot as plt
 
mean = 0; std = 1; variance = np.square(std)
x = np.arange(-5,5,.01)
f = np.exp(-np.square(x-mean)/2*variance)/(np.sqrt(2*np.pi*variance))
 
plt.plot(x,f)
plt.ylabel('gaussian distribution')
plt.show()

With libs:

import matplotlib.pyplot as plt
import numpy as np
import scipy.stats as stats
import math
 
mu = 0
variance = 1
sigma = math.sqrt(variance)
x = np.linspace(mu - 3*sigma, mu + 3*sigma, 100)
plt.plot(x, stats.norm.pdf(x, mu, sigma))  # probability density function
plt.show()