eng Science and Education Publishing American Journal of Applied Mathematics and Statistics 2328-7292 2017-07-05 5 2 49 53 10.12691/ajams-5-2-2 AJAMS2017522 article Matthew Chukwuma Michael megawaves4life@yahoo.com 1 Oyeka Cyprain Anene 2 Ashinze Mpuruoma Akudo 2 Igabari John Nwabueze 3 Department of Mathematics and Statistics, School of Applied Sciences, Delta State Polytechnic, Ogwashi-Uku Department of Statistics, Faculty of Physical Sciences, Nnamdi Azikiwe University, Awka, Anambra State Department of Mathematics, Delta State University, Abraka, Delta State This paper seeks to develop a generalized method of generating the moments of random variables and their probability distributions. The Generalized Moment Generating Function is developed from the existing theory of moment generating function as the expected value of powers of the exponential constant. The methods were illustrated with the Beta and Gamma Family of Distributions and the Normal Distribution. The methods were found to be able to generate moments of powers of random variables enabling the generation of moments of not only integer powers but also real positive and negative powers. Unlike the traditional moment generating function, the generalized moment generating function has the ability to generate central moments and always exists for all continuous distribution but has not been developed for any discrete distribution. http://pubs.sciepub.com/ajams/5/2/2/ajams-5-2-2.pdf generalized moments generating functions distribution function arbitrary constant