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Research Article Open access CC BY 4.0

Generalized Moment Generating Functions for Some Continuous Multivariate Probability Distributions

Matthew Chukwuma Michael

Asian Journal of Probability and Statistics · pp. 54–68 · Published 10 Sep 2022

10.9734/ajpas/2022/v19i430477

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.

Generalized moments generating function multivariate probability distributions multivariate normal distribution multivariate gamma distribution Dirrichlet distribution

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