Continuous counterparts of Poisson and binomial distributions and their properties

TL;DR

This paper introduces continuous analogs of Poisson and binomial distributions using integral representations, explores their properties, and establishes convergence and relationships with classical functions and processes.

Contribution

It presents novel continuous counterparts of discrete distributions based on integral representations and analyzes their properties and convergence behavior.

Findings

Continuous binomial converges weakly to continuous Poisson

Connections established with Volterra functions and mma-process

Integral representations facilitate new distribution analysis

Abstract

On the basis of integral representations of Poisson and binomial distribution functions via complete and incomplete Euler \Gamma- and B-functions, we introduce and discuss continuous counterparts of the Poisson and binomial distributions. The former turns out to be closely related to classical Volterra functions as well. Under usual conditions, we also prove that the sequence of continuous binomial distributions converges weakly to the continuous Poisson one. At the end, we discuss a relationship between the continuous Poisson distribution and the \Gamma-process.