<Tr> <Th> MGF </Th> <Td> exp (μ ′ t + 1 2 t ′ Σ t) (\ displaystyle \ exp \! (\ Big () (\ boldsymbol (\ mu))' \ mathbf (t) + (\ tfrac (1) (2)) \ mathbf (t)' (\ boldsymbol (\ Sigma)) \ mathbf (t) (\ Big))) </Td> </Tr> <Tr> <Th> CF </Th> <Td> exp (i μ ′ t − 1 2 t ′ Σ t) (\ displaystyle \ exp \! (\ Big () i (\ boldsymbol (\ mu))' \ mathbf (t) - (\ tfrac (1) (2)) \ mathbf (t)' (\ boldsymbol (\ Sigma)) \ mathbf (t) (\ Big))) </Td> </Tr> <P> In probability theory and statistics, the multivariate normal distribution or multivariate Gaussian 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 . </P> <P> The multivariate normal distribution of a k - dimensional random vector X = (X, X,..., X) can be written in the following notation: </P>

Probability density function of a multivariate gaussian distribution
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