A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

# A Generalized Geometric Distribution from Vertically Dependent Bernoulli Random Variables

## Abstract

This paper generalizes the notion of the geometric distribution to allow for dependent Bernoulli trials generated from dependency generators as defined in Traylor and Hathcock’s previous work. The generalized geometric distribution describes a random variable $X$ that counts the number of dependent Bernoulli trials until the first success. The main result of the paper is  $X$ can count dependent Bernoulli trials from any dependency structure and retain the same distribution. That is, if  $X$ counts Bernoulli trials with dependency generated by $\alpha_{1} \in \mathcal{C}_{\delta}$, and $Y$ counts Bernoulli trials with dependency generated by $\alpha_{2} \in \mathscr{C}_{\delta}$, then the distributions of  $X$ and $Y$ are the same, namely the generalized geometric distribution. Other characterizations and properties of the generalized geometric distribution are given, including the MGF, mean, variance, skew, and entropy.

## Introduction

The standard geometric distribution counts one of two phenomena:

1.  The count of i.i.d. Bernoulli trials until the first success
2.  The count of i.i.d. Bernoulli trials that resulted in a failure prior to the first success

The latter case is simply a shifted version of the former. However, this distribution, in both forms, has limitations because it requires a sequence of independent and identically distributed Bernoulli trials. Korzeniowski [2] originally defined what is now known as first-kind (FK) dependent Bernoulli random variables, and gave a generalized binomial distribution that allowed for dependence among the Bernoulli trials. Traylor [4] extended the work of Korzeniowski into FK-dependent categorical random variables and derived a generalized multinomial distribution in a similar fashion. Traylor and Hathcock [5] extended the notion of dependence among categorical random variables to include other kinds of dependency besides FK dependence, such as sequential dependence. Their work created a class of vertical dependency structures generated by a set of functions

$$\mathscr{C}_{\delta} = \{\alpha: \mathbb{N}_{\geq 2} \to \mathbb{N} : \alpha(n) < n \text{ and } \forall n \exists j \in \{1,\ldots,n-1\} : \alpha(n) = j\},$$

where the latter property is known as dependency continuity. In this paper, we derive a generalized geometric distribution from identically distributed but dependent Bernoulli random variables. The main result is that the pdf for the generalized geometric distribution is the same regardless of the dependency structure. That is, for any $\alpha \in \mathscr{C}_{\delta}$ that generates a sequence of identically distributed but dependent Bernoulli trials, the generalized geometric distribution remains unchanged.