A multivariate Gaussian distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. A random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution.

Before looking at the multivariate Gaussian distribution, let's consider the univariate distribution.

A univariate distribution is generated with the following formula:

In the preceding formula, the following applies:

• σ represents the standard deviation of the distribution
• µ represents the mean of the distribution

Given the preceding two parameters, a Gaussian distribution with a certain mean and standard deviation is generated by varying the values of x from -∞ to ∞.

A typical plot of a Gaussian distribution for different values of mean and standard deviation can look as follows:

A multivariate gaussian distribution is very similar to a univariate Gaussian distribution. However, given that there are multiple variables involved, a univariate mean (the mean of a single variable) is replaced by a multivariate mean (the mean of each of the variables), and the variance of a single variable is replaced by the covariance of the multiple variables present in a multivariate Gaussian distribution.

Thus, the formula of a multivariate Gaussian distribution is as follows:

A typical chart of a bivariate normal distribution for a given value of x can look like the following:

A k-variate multinormal distribution is a generalization of the bivariate normal distribution. The k-multivariate distribution with mean vector µ and covariance matrix ∑ is denoted by the following:

In the following section, we will look at calculating the probability associated with a multivariate Gaussian distribution.