Generalized Moment Generating Functions of Random Variables and Their Probability Density Functions (original) (raw)
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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.
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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
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
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.
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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.
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.
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