This tag is for questions on the Gaussian, or normal probability distribution, which may include multi-dimensional normal distribution. The normal distribution is the most common type of distribution assumed in technical stock market analysis and in other types of statistical analyses.

Normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Graphically, normal distribution will appear as a bell curve.

However, many other distributions are bell-shaped (such as the Cauchy, Student's $~t-~$, and logistic distributions).

**Probability density function:** The general formula for the probability density function of the normal distribution is
\begin{equation*}
P(x)=\frac{1}{\sigma\sqrt{2\pi}}e^{-~\frac{(x-\mu)^2}{(2\sigma^2)}},~x\in (-\infty,\infty).
\end{equation*}

where

- $~\mu~$ is the
**mean**or**expectation**of the distribution (and also its median and mode), - $~\sigma~$ is the
**standard deviation**, and - $~\sigma ^{2}~$ is the
**variance**.

The case where $~μ = 0~$ and $~σ = 1~$ is called the **standard normal distribution**. The equation for the standard normal distribution is

\begin{equation*} P(x)=\frac{1}{\sqrt{2\pi}}e^{-~\frac{x^2}{2}},~x\in (-\infty,\infty). \end{equation*}

Since the general form of probability functions can be expressed in terms of the standard distribution, all subsequent formulas in this section are given for the standard form of the function.

**Applications:** The normal distribution is useful because of the central limit theorem. In its most general form, under some conditions (which include finite variance), it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become normally distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes (such as measurement errors) often have distributions that are nearly normal. Moreover, many results and methods (such as propagation of uncertainty and least squares parameter fitting) can be derived analytically in explicit form when the relevant variables are normally distributed.

**References:**