Normal distribution

A normal or Gaussian distribution has the following form

p(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left( \frac{(x-\mu)^2}{2\sigma^2}\right).

Where $\mu$ is the mean and $\sigma$ is the standard deviation. The normal distribution occurs almost everywhere. Most other distributions such as the binomial and Poisson distributions approach a normal distribution when the sample size is large.

The cumulative distribution function for a normal distribution is

\begin{align*} P(x) &= \int_{-\infty}^x \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left( \frac{(y-\mu)^2}{2\sigma^2}\right) \d y \\ &= 1 - \frac12 \erfc\left(\frac{x-\mu}{\sqrt 2\sigma} \right). \end{align*}