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