A New Generalized Poisson Distribution And Its Application

The Poisson distribution which was frst formulated in1837by the famousFrench mathematician and physicist Simeon Denis Poisson. It is an importantdiscrete stochastic variable. Poisson process and compound Poisson process playimportant roles in many felds, such as physics, fnancial, economics, engineeringand so on. Many scholars have done a lot of work on Poisson sheet, generalizedPoisson sheet, compound Poisson sheet. But they mainly extended their timeparameters and have not considered whether the distribution function itself canbe generalized.In this article, we try to extend the structure form of the Poisson distri-bution. we defne a new generalized Poisson distribution, give its mathematicalexpectation, variance and characteristic function; we also defne a new general-ized Poisson process, and give some simple study about its basic properties, thenumerical characteristics, the time of arrival; fnally, we discuss two problems inthe study of the new generalized Poisson process.This thesis is divided into fve chapters. In chapter one, we introduce theapplication background and present situation of research, and generalize the con-tent of this thesis and the main results, and also give the notations, defnitions,needed results. In chapter two, we defne a new generalized Poisson distribution,and obtain a computational formula for the constant Hm(λ) to be confrmed inthe new generalized Poisson distribution through the ideas of diferential equa-tion. In chapter three, we defne a new generalized Poisson process, and alsostudy its properties. In chapter four, we discuss two problems in the study ofthe new generalized Poisson process, and give detailed proofs for all results ofresearch. In chapter fve, we conclude the whole thesis and point out some direc-tions for further investigation.