# Normal Distribution 

##### Definition

The normal (or Gaussian) distribution is a very common continuous probability distribution in probability theory.

Parameters
Notation \mathcal N(\mu, \sigma^2), \mu \in \R, \sigma \gt 0
Mean \mu
Variance \sigma^2
PDF f_X (x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}, x \in \R

## Probability Density

f_X (x) = \frac{1}{\sqrt{2 \pi \sigma^2}} e^{-\frac{(x-\mu)^2}{2 \sigma^2}}

with the mean \mu, the standard deviation \sigma.

## Properties

• It is symmetric around the point x=\mu which is at the same time the mode, the median and the mean of the distribution.
• The area under the curve and over the x-axis is equal to one.