A probabilistic interpretation of the Bell polynomials

Abstract

In this paper, we obtain a probabilistic relationship between the exponential Bell polynomials and the weighted sums of independent Poisson random variables. A recently established probabilistic connection between the Adomian polynomials and independent Poisson random variables can be derived from the obtained relationship. This result has importance because any known identity for the exponential Bell polynomials will generate a new identity for the Poisson random variables. We use the obtained relationship to derive several new identities for the joint distribution of weighted sums of independent Poisson random variables. Few examples are provided that substantiate the obtained identities.

Keywords:

• Bell polynomials
• Poisson distribution of order n
• probabilistic method
• recurrence relations
• Adomian polynomials

AMS CLASSIFICATION:

• 60E05
• 05A19

Acknowledgement

The authors thank the anonymous reviewers for their comments and suggestions which led to improvements.