For any real numbers $\mu$ and $\sigma > 0$ , the Gaussian probability distribution function with mean $\mu$ and variance $\sigma^2$ is defined by $$ f(x) = \frac{1}{\sqrt{2 \pi \sigma^2}} \exp \left( - \tfrac12 \left( \tfrac{x - \mu}{\sigma} \right) ^2 \right). $$ When $\mu=0$ and $\sigma = 1$ , it is usually called standard normal distribution.

A random variable $X$ having distribution density $f$ is said to be a normally distributed random variable, denoted by $X\sim N(\mu,\sigma^2)$ . It has expected value $\mu$ , and variance $\sigma^2$ .

Cumulative distribution function

The cumulative distribution function of a standard normal variable, often denoted by $$ \Phi(z) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^z e^{-x^2/2} \, dx\,, $$ cannot be calculated in closed form in terms of the elementary functions, but its values are tabulated in most statistics books and here, and can be computed using most computer statistical packages and spreadsheets.

Uses of the Gaussian distribution

The normal distribution is probably the most frequently used distribution. Its graph looks like a bell-shaped function, which is why it is often called bell distribution.

The normal distribution is important in probability theory and statistics. Empircally, many observed distributions, such as of people's heights, test scores, experimental errors, are found to be more or less to be Gaussian. And theoretically, the normal distribution arises as a limiting distribution of averages of large numbers of samples, justified by the central limit theorem.

Figure 1: Graph of densities of the normal distribution for various values of the standard deviation $\sigma$

Properties

• If $Z$ is a standard normal random variable, then $X = \sigma Z + \mu$ is distributed as $N(\mu, \sigma^2)$ , and conversely.
• The sum of any finite number of independent normal variables is itself a normal random variable.

Relations to other distributions

1. The standard normal distribution can be considered as a Student-t distribution with infinite degrees of freedom.
2. The square of the standard normal random variable is the chi-squared random variable of degree 1. Therefore, the sum of squares of $n$ independent standard normal random variables is the chi-squared random variable of degree $n$ .