A Gaussian distribution is one way of distributing random numbers. This comes up a lot in everyday stuff. This is because the central limit theorem says the sums of many independent random variables converge to a Gaussian.

A Gaussian is any function of the form

f(x)=A e−(x−x0)2/2σ2f(x) = A\, e^{-(x-x_0)^2 / 2\sigma^2}

The probability distribution normalizes it to 11

P(x)=12πσ2 e−(x−μ)22σ2P(x) = \frac{1}{\sqrt{2\pi\sigma^2}}\,e^{-\frac{(x-\mu)^2}{2\sigma^2}}

Properties:

  1. Symmetric about μ\mu
  2. Note
E[x]=μE[(x−μ)2]=σ2\mathbb{E}[x]=\mu\quad \mathbb{E}[(x-\mu)^2]=\sigma^2