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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.

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