Multivariate normal distribution

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In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional (univariate) normal distribution to higher dimensions. One definition is that a random vector is said to be k-variate normally distributed if every linear combination of its k components has a univariate normal distribution. Its importance derives mainly from the multivariate central limit theorem. The multivariate normal distribution is often used to describe, at least approximately, any set of (possibly) correlated real-valued random variables each of which clusters around a mean value. The multivariate normal distribution of a k-dimensional random vector can be written in the following notation: or to make it explicitly known that X is k-dimensional, with k-dimensional mean vector and covariance matrix such that and . The inverse of the covariance matrix is called the precision matrix, denoted by . A real random vector is called a standard normal random vector if all of its components are independent and each is a zero-mean unit-variance normally distributed random variable, i.e. if for all . A real random vector is called a centered normal random vector if there exists a deterministic matrix such that has the same distribution as where is a standard normal random vector with components. A real random vector is called a normal random vector if there exists a random -vector , which is a standard normal random vector, a -vector , and a matrix , such that . Formally: Here the covariance matrix is . In the degenerate case where the covariance matrix is singular, the corresponding distribution has no density; see the section below for details. This case arises frequently in statistics; for example, in the distribution of the vector of residuals in the ordinary least squares regression. The are in general not independent; they can be seen as the result of applying the matrix to a collection of independent Gaussian variables .

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