A New Generalized Moment Generating Function of Random Variables (original) (raw)
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. Cite This Article: Matthew Chukwuma Michael, Oyeka Cyprain Anene, and Ashinze Mpuruoma Akudo, " Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions. "
The moment generating function of the distribution of X^c Y^ d
This paper presents a proposed alternative method of expression of joint distribution of powers of two continuous random variables when both powers are necessarily not whole numbers. Hence, in finding the alternative moment generating function (AMGF) of the joint distribution of some functions of a given random variable, it is not necessary to find and use the joint distribution of these functions, it is sufficient to simply use the joint distribution of the two random variables and hence quicker to use. Unlike the regular moment generating functions, the alternative method of moment generating function is known to always exist for all continuous probability distributions.
Applying the Moment Generating Functions to the Study of Probability Distributions
Informatica Economica, 2007
In this paper, we describe a tool to aid in proving theorems about random variables, called the moment generating function, which converts problems about probabilities and expectations into problems from calculus about function values and derivates. We show how the moment generating function determinates the moments and how the moments can be used to recover the moment generating function. Using of moment generating functions to find distributions of functions of random variables is presented. A standard form of the central limit theorem is also stated and proved.
SSRN Electronic Journal, 2019
Statistics have been widely used in many disciplines including science, social science, business, engineering, and many others. One of the most important areas in statistics is to study the properties of distribution functions. To bridge the gap in the literature, this paper presents the theory of some important distribution functions and their moment generating functions. We introduce two approaches to derive the expectations and variances for all the distribution functions being studied in our paper and discuss the advantages and disadvantages of each approach in our paper. In addition, we display the diagrams of the probability mass function, probability density function, and cumulative distribution function for each distribution function being investigated in this paper. Furthermore, we review the applications of the theory discussed and developed in this paper to decision sciences.
2017
The main objective of this article is to show the derivation of the moment generating function of the four-parameter of generalized (G4F) F distribution. Through parameterization of its moment generating function, the behavior in relation to several well-known generalized distributions is presented. By utilizing MacLaurin series expansion and Stirling formula, it is shown that with parameterization of its moment generating function, the generalized F distribution might have special relationship to several well-known generalized distributions, such as generalized beta of the second kind (GB2), generalized log-logistic (G4LL), and generalized gamma (G3G) distributions. Keywords: Moment generating function; Generalized beta of the second kind distribution; Generalized log-logistic distribution; Generalized gamma; MacLaurin series; Stirling formula
Discrete distributions from moment generating function
Applied Mathematics and Computation, 2006
The recovering of positive discrete distributions from their moment generating function (mgf) is considered. From mgf 8 and some integer moments, proper fractional moments are obtained. The latter represent the available information of the 9 distribution. Then maximum entropy machinery is invoked to find the approximate distribution. It is proved that the 10 approximant converges in entropy, in information divergence and then in total variation, so that accurate expected values 11 may be obtained. Some numerical experiments are illustrated. 12
Moment-Generating Functions and Reproductive Properties of Distributions
In this paper, an important mathematical concept which has many applications to the probabilistic models are presented. Some of the important applications of the moment-generating function to the theory of probability are discussed. Each probability distribution has a unique moment-generating function, which means they are especially useful for solving problems like finding the distribution for sums of random variables. Reproductive properties of probability distributions with illustrated examples are also described.
A Note on Moment Generating Function of a Linear Combination of Order Statistics from a
2013
In this note we propose an extended skew-Laplace distribution. We obtain explicit expressions for moment generating function and the two first moments of this distribution. Next, we show that the distribution of a linear combination of order statistics from a bivariate Laplace distribution can be expressed as a mixture of extended skew-Laplace distributions. This mixture representation enables us to derive moment generating function and moments of this linear combination.
Moment generating function of the generalized α - μ distribution with applications
IEEE Communications Letters, 2000
In this letter, we consider the α−μ channel fading model and we evaluate the moment generating function (MGF) for the probability density function characterizing this new channel model. The derived MGF expression is used in evaluating the bit error rate for different coherent modulation techniques over this generalized fading channel. We also derive an expression for the outage probability for this channel model. All the derived expressions are in closed forms and general that can reduce to the well known fading channel distributions in the literature such as Rayleigh, Nakagami-m, and Weibull model as special cases.
Moment identity for discrete random variable and its applications
Statistics, 2012
In this paper, we obtain a moment identity applicable to a general class of discrete probability distributions. We then derive the corresponding identities for modified power series, Ord and Katz families. It is noted that the proposed identity has potential applications in different fields.
Possibilistic moment generating functions
Applied Mathematics Letters, 2011
Following Carlsson and Fuller (2001) [2], recently Thavaneswaran et al. (2009) [1] have introduced higher order weighted possibilistic moments of fuzzy numbers. In this paper, we define the weighted possibilistic moment generating functions (MGF) of fuzzy numbers and obtain the closed form expressions for triangular, trapezoidal and parabolic fuzzy numbers. Applications involve derivation of higher order possibilistic moments of volatility models (see Thavaneswaran et al. (2009) [1] for details).
On Generalized Moment Exponential Distribution and Power Series Distribution
2020
In this research paper, a new life time family is introduced. Sadaf [1] proposed a moment exponential power series (MEPS) distribution. Generalized moment exponential power series (GMEPS) distribution is a general form of MEPS distribution. It is characterized by compounding GME distribution and power series (PS) distribution. This new family has some new sub models such as GME geometric distribution, GME Poisson (GMEP) distribution, GME logarithmic (GMEL) distribution and GME binomial (GMEB) distribution. We provide statistical properties of GMEPS family of distributions. We find here expression of quantile function based on Lambert W function, the density function of rth order statistic and moments of GMEPS distribution. Descriptive expressions of Shannon entropy and Rényi entropy of new general model are found. We provide special sub-models of the GMEPS family of distributions. The maximum likelihood (ML) estimation method is used to find estimates of the parameters of GMEPS distribution. Simulation study is carried out to check the convergence of new estimators. We apply GMEPS family of distributions on two sets of real data. Original Research Article Iqbal et al.; AJPAS, 6(1): 1-21, 2020; Article no.AJPAS.53280 2
A Generalized Transmuted Moment Exponential Distribution: Properties and Application
Academic Journal of Applied Mathematical Sciences, 2019
This paper introduces a new generalization of moment exponential (or length biased) distribution. The new model is referred to as generalized transmuted moment exponential distribution. This model contains some new existing distributions. Structural properties of the suggested distribution including closed forms for ordinary and incomplete moments, quantile and generating functions and Rényi entropy are derived. Maximum likelihood estimation is employed to obtain the parameter estimators of the new distribution. We illustrate the importance of the new model by means of three applications to real data sets.
This article introduces a new family of lifetime distributions called the exponentiated moment exponential power series (EMEPS) which generalizes the moment exponential power series (MEPS) proposed by Sadaf (2013). This new family is obtained by compounding the exponentiated moment exponential and truncated power series distributions, where the compounding procedure follows same way that was previously carried out by Adamidis and Loukas (1998). The new family contains some new distributions such as exponentiated moment exponential geometric distribution, exponentiated moment exponential Poisson distribution, exponentiated moment exponential logarithmic distribution and exponentiated moment exponential binomial distribution. Some former works derived by Sadaf 2014 such as moment exponential geometric and moment exponential Poisson distributions are special cases of the new EMEPS family. We obtain several properties of EMEPS family, among them; quantile function, order statistics, moments and entropy. Some special models in the exponentiated moment exponential power series family of distributions are provided. Maximum likelihood (ML) method is applied to obtain parameter estimates of the EMEPS family. A simulation study is carried out to check the consistency of the ML estimators of the parameters. Two real data sets are used to validate the distributions and the results demonstrate that the sub-models from the family can be considered as suitable models under several real situations.
Characteristics of the Moment Generating Function of Generalized Gamma Distribution
2013
In this paper, some characteristics of Generalized Gamma distribution and Gamma distribution as a special case of the generalized Gamma distribution are described. Generalized Gamma distribution (GG()) is a continuous probability distribution with three parameters. GG distribution and Gamma distribution has the same domain for a non-negative x. In addition to the parameters and the domain, GG distribution has some characteristics such as probability density function, expected value, variance, and moment generating function. In the approximated GG distribution to the Gamma distribution through the moment generating function, used the Maclaurin series. This paper also presents a graph of the probability density function of GG disrtribution and Gamma distribution separately, for each of its parameters. In graphing, used the R version 3.0.1 program.
Onk-Gamma andk-Beta Distributions and Moment Generating Functions
Journal of Probability and Statistics, 2014
The main objective of the present paper is to definek-gamma andk-beta distributions and moments generating function for the said distributions in terms of a new parameterk>0. Also, the authors prove some properties of these newly defined distributions.
A generalized moments expansion
Physics Letters A, 2006
A new generalized moments expansion (GMX), which is based on the t-expansion, is derived. The well-known connected moments expansion (CMX) and alternate moments expansion (AMX) are shown to be special cases of the GMX.
Generalized Probability Functions
Advances in Mathematical Physics, 2009
From the integration of nonsymmetrical hyperboles, a one-parameter generalization of the logarithmic function is obtained. Inverting this function, one obtains the generalized exponential function. Motivated by the mathematical curiosity, we show that these generalized functions are suitable to generalize some probability density functions pdfs . A very reliable rank distribution can be conveniently described by the generalized exponential function. Finally, we turn the attention to the generalization of one-and two-tail stretched exponential functions. We obtain, as particular cases, the generalized error function, the Zipf-Mandelbrot pdf, the generalized Gaussian and Laplace pdf. Their cumulative functions and moments were also obtained analytically.
A new family of distributions based on probability generating functions
Sankhya B, 2011
This paper examines a method for generating new classes of distributions which arise naturally in practice. The generated classes of distributions include the well known Marshall and Olkin class of distributions and can be thought of as mixing two discrete distributions or a discrete distribution with an absolutely continuous distribution. Properties of these classes of distributions are derived and a number of existing results in the literature are recovered as special cases. Finally, failure rates for a special class of distributions which are obtained when the discrete distribution is assumed to have a Harris form are given.
DISTRIBUTION CHARACTERIZATION IN A PRACTICAL MOMENT PROBLEM
2004
We investigate a problem connected with the evaluation of the asymptotic probability distribution function (APDFs) given from a set of finite order moments by applying the Gram-Schmidt process with the aid of computer algebra. By selecting weighting (discrete or continuous) function of similar shape to desired (APDFs), orthogonal polynomial series are obtained that are stable at high order and allow accurate approximation of tail probabilities.