Here we directly evaluate an -dimensional integral

as an integral representation of Wishart multivariate gamma function  by deriving the limits of integration for each and every distinct entry of matrix  With our formulae, one can readily simulate a correlation matrix, and indeed any positive-definite matrix. We also derive two marginal (on the  entries of the correlation matrix and on the m variances, respectively) probability density functions (that seem new, except for the case of  of Wishart probability density function.

Wishart multivariate gamma function, integral representation, a direct evaluation, marginal probability density functions, Wishart probability density function.