Abstract
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.
Highlights
We propose in this paper to develop a more versatile, easier and quicker to apply function which for lack of better nomenclature shall be called the generalized moment generating function
This paper has developed and presented the generalized moment generating functions of random variables and their probability distributions
The method has been shown to be quicker and easier to apply than the traditional moment generating functions which may not exist for some distributions
Summary
Suppose ff(xx) = 2xx for the random variable, XX; 0 < xx < 1. For cc = 1, nn = 1, the first moment of XX about λλ for the distribution is. The variance of the distribution of X 2 is 2. The first moment of the random variable, XX, about λλ. Equation 8 gives the variance of the random variable, XX for the distribution 2xx; 0 < xx < 1. The same result would be obtained using classical methods. If λλ = 0 , the first non-central moment of the distribution of X 2 becomes,.
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