Some Theories Of The Class Of Generalized Beta Distributions

In practice, many proplems are often too complicated to be modeled by well-known basic probability distributions. Consequently, we need to develop generalized probability distributions that can handle these complex situations.In this dissertation, we considered the family of generalized Beta distributions, proposed the family of generalized multivariate Beta distributions, and obtained some results of generalized multivariate Beta distributions. The paper is organized as follows:In Chapter 2, an explicit solution of a Jacobian, which had been considered by Robb J. Muirhead in [50] , where its implicit solution was given via exterior differential, had been calculated.In Chapter 3, a generalization of Vandermonde determinant had been calculated, then, by this result an identity was proved and generalized, which is of great use for stochastic process, and the distribution of random vector of the sum of independent different random variables were discussed.In Chapter 4, some examples of generalized Beta distribution had been given; In Chapter 5, the moment generating function of multivariate Beta distribution had been derived, the definition of generalized multivariate Beta distribution had been proposed, and the marginal distribution and conditional distribution of generalized multivariate Beta distribution had been derived; In Chapter 6, the definition of generalized Dirichlet distribution had been proposed and some properties had been obtained.In Chapter 7, the class of generalized Beta exponential distributions had been studied, and he moments, moment generating function, characteristic function, the extremum of density function and some others properties had been investigated.In Chapter 8, the class of generalized Beta Weibull distributions had been simply discussed.Finally, some appendixes of this paper were given.