Independence of Linear Statistics with Random Coefficients and Characterizations of Geometric and Poisson Distributions

TL;DR

This paper characterizes geometric and Poisson distributions through the independence of linear forms with random coefficients, extending known results from exponential distributions to discrete cases.

Contribution

It introduces a novel characterization of geometric and Poisson distributions based on the independence of linear forms with random coefficients, providing a discrete analog to existing exponential distribution theorems.

Findings

Characterization of geometric distribution via independence of linear forms with random coefficients

Extension of exponential distribution characterization to discrete distributions

Identification of Poisson law through linear statistics independence

Abstract

There is given a characterization of the geometric distribution by the independence of linear forms with random coefficients. The result is a discrete analog of the corresponding theorem on exponential distribution. The property of linear statistics independence is also a characterization of Poisson law. Keywords: geometric distribution; exponential distribution; Poisson distribution; linear forms; random coefficients