Normal distribution

probability density function

cumulative probability function

The normal distribution is a key distribution in the field of probability. It is also known as the the Gaussian distribution, after mathematician Carl Gauss, and the bell curve. The normal probability density function (PDF) is

<math>

f(x)=\frac{1}{\sqrt{2\pi\sigma^2}} \exp\left[-\frac{1}{2\sigma^2}\left(x-\mu\right)^2\right] </math> where <math>\mu</math> is the mean and <math>\sigma^2</math> is the variance.

If <math>\mu=0</math> and <math>\sigma=1</math>, the distribution is called the standard normal distribution, often denoted by <math>\phi</math>:

<math>

\phi(x) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{1}{2} x^2). </math>