Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions

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.


Introduction
The ℎ moment of a random variable, , about an arbitrarily chosen constant, , is defined as the expected value of the ℎ power of the difference between the random variable, , and the arbitrarily chosen constant, . If is equal to zero the moment of interest is called the non-central moment or moment about zero; however, if the constant, , is equal to , the mean of the distribution, interest is on the central moments [1,2,3].
The basic method of determining the ℎ non-central moment of a random variable, , is the moment generating function (here referred to as the traditional moment generating function), defined as ( ) = ( ). [4,5].
( ) is usually difficult to evaluate and may not exist for some distributions.
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. The method, which is similar to the traditional moment generating function in formation is able to generate moments of powers of random variables that are not necessarily positive integers but may be any real number.

The Generalized Moment Generating Function
Let ba a random variable whose is denoted by ( ); is any real number that does not need to be positive or integral and is any arbitrarily chosen constant. Let ( ) define the generalized moment generating function of . Then, , may be evaluated as follows: The coefficient of ! In Equation 2 gives the ℎ generalized moment of . That is, Equation 3 yields the ℎ generalized moment of the random variable, . ( ) is the non-central moment or the generalized moment about zero of the random variable, .
The first moment of the random variable, , about . Now, if = 0, the first non-central moment becomes The second generalized moment may be obtained using Equation 4 as ( ) 2 2 4 1 1; .
Hence for 2 , 3 Equation 8 gives the variance of the random variable, for the distribution 2 ; 0 < < 1. The same result would be obtained using classical methods.
Hence the first generalized moment of the distribution Now if = 0 , the first non-central moment of the distribution of 1 2 X becomes, The second generalized moment of the distribution of Hence the variance of the distribution of 1 2 X is 2. .
Hence, the first generalized moment is given as Thus if = 0, the first non-central moment becomes Also, the second generalized moment of the distribution is Hence the second central moment is

The Beta Family of Distributions
A random variable is said to have a beta distribution if its density function is of the form: . Γ Γ n n r n r cr n G c r cr .
If = 1, interest is on the first moment of about then it yields ( ) That is, the variance of the Beta distribution.

The Gamma Family of Distributions
Let be a gamma random variable with the density function: integrating, simplifying and taking the coefficient of ! n t n yields . Γ For the second generalized moment of the where has the gamma distribution is The generalized moment generating function can be used to obtain moments of powers of random variables with non-integer negative indices.

The Normal Distribution
The generalized moment generating function of the random where has the normal distribution, with parameters and 2 , and with given by  implying that the distribution is symmetric (Arua et al 1997).
The kurtosis of the normal distribution may be obtained as

Conclusion
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. Thus, the generalized moment generating function is more versatile than the traditional moment generating function. The new method was illustrated with a general probability distribution function, the beta family of distributions, the gamma family of distributions and the normal distribution. However, this method has not been developed for discrete probability distributions.