Generalized Moment Generating Functions for Some Continuous Multivariate Probability Distributions

Abstract

The traditional moment generating functions of random variables and their probability distributions are known to not exist for all distributions and/or at all points and, where they exist, serious difficult and tedious manipulations are needed for the evaluation of higher central and non-central moments. This paper developed the generalized multivariate moment generating function for some random vectors/matrices and their probability distribution functions with the intention to replace the traditional/conventional moment generating functions due to their simplicity and versatility. The new functions were developed for the multivariate gamma family of distributions, the multivariate normal and the dirrichlet distributions as a binomial expansion of the expected value of an exponent of a random vector/matrix about an arbitrarily chosen constant. The functions were used to generate moments of random vectors/matrices and their probability distribution functions and the results obtained were compared with those from existing traditional/conventional methods. It was observed that the functions generated same results as the traditional/conventional methods; in addition, they generated both central and non-central moments in the same simple way without requiring further tedious manipulations; they gave more information about the distributions, for instance while the traditional method gives skewness and kurtosis values of  and  respectively for -variate multivariate normal distribution, the new methods gives ((0))p*1  and  respectively and; they could generate moments of integral and real powers of random vectors/matrices.