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### Enterprise Risk Management Formula Book Appendix A.5: Probability Distributions: Multivariate probability distributions

Multivariate normal (i.e. Gaussian) distribution

The multivariate probability distribution  where  is a vector of  elements and  is an  non-negative definite matrix has the following joint density function (where  is the determinant of V)

The means of the individual marginal distributions are  where  and the covariance between the ’th and the ’th marginal distributions are  where the  are the elements of . Its moment generating function is  and its characteristic function is . The multivariate normal distribution has as its copula the Gaussian copula.

A bivariate random variable  follows a standard bivariate normal distribution if it has  and . More generally, a multivariate normal distribution is a standard multivariate normal distribution if  and a covariance matrix which is also a correlation matrix, i.e. where the variance of each individual marginal distribution is 1.

For numerical values of the cumulative distribution function of the standard bivariate normal distribution see here.