Continuous counterparts of Poisson and binomial distributions and their properties

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.