Type $1$, $2$, $3$ and $4$ $q$-negative binomial distribution of order $k$

TL;DR

This paper derives exact formulas for various types of $q$-negative binomial distributions of order $k$, considering different run enumeration schemes and geometrically varying success probabilities in binary trials.

Contribution

It introduces four new types of $q$-negative binomial distributions of order $k$ with exact formulas for complex run scenarios and variable success probabilities.

Findings

Derived explicit formulas for four types of $q$-negative binomial distributions.

Analyzed distributions for overlapping and non-overlapping success runs.

Extended distribution models to geometrically varying success probabilities.

Abstract

We study the distributions of waiting times in variations of the negative binomial distribution of order $k$. One variation apply different enumeration scheme on the runs of successes. Another case considers binary trials for which the probability of ones is geometrically varying. We investigate the exact distribution of the waiting time for the $r$-th occurrence of success run of a specified length (non-overlapping, overlapping, at least, exactly, $\ell$-overlapping) in a $q$-sequence of binary trials. The main theorems are Type $1$, $2$, $3$ and $4$ $q$-negative binomial distribution of order $k$ and $q$-negative binomial distribution of order $k$ in the $\ell$-overlapping case. In the present work, we consider a sequence of independent binary zero and one trials with not necessarily identical distribution with the probability of ones varying according to a geometric rule. Exact formulae for the distributions obtained by means of enumerative combinatorics.