Symmetric generalized binomial distributions

TL;DR

This paper introduces a new symmetric generalization of the binomial distribution based on generating functions, maintaining the symmetry between win and loss probabilities while ensuring non-negativity constraints.

Contribution

It proposes a novel symmetric binomial distribution generalization using generating functions, expanding previous asymmetric models.

Findings

Maintains symmetry between win and loss probabilities.

Ensures non-negativity constraints in the distribution.

Provides a mathematical framework for symmetric generalization.

Abstract

In two recent articles we have examined a generalization of the binomial distribution associated with a sequence of positive numbers, involving asymmetric expressions of probabilities that break the symmetry {\it win-loss}. We present in this article another generalization (always associated with a sequence of positive numbers) that preserves the symmetry {\it win-loss}. This approach is also based on generating functions and presents constraints of non-negativeness, similar to those encountered in our previous articles.