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

Jungtaek Oh

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.

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

Abstract

We study the distributions of waiting times in variations of the negative binomial distribution of order . 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 -th occurrence of success run of a specified length (non-overlapping, overlapping, at least, exactly, -overlapping) in a -sequence of binary trials. The main theorems are Type , , and -negative binomial distribution of order and -negative binomial distribution of order in the -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.
Paper Structure (7 sections, 15 theorems, 154 equations, 1 figure)

This paper contains 7 sections, 15 theorems, 154 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Terminology and notation
  3. Type $I$ $q$-negative binomial distribution of order $k$
  4. Type $II$ $q$-negative binomial distribution of order $k$
  5. Type $III$ $q$-negative binomial distribution of order $k$
  6. Type $IV$ $q$-negative binomial distribution of order $k$
  7. $q$-negative binomial distribution of order $k$ in the $\ell$-overlapping case

Key Result

Lemma 3.1

[yalcin2014q] For $0<q\leq1$, $A_{q}^{k}(r,s,t)$ obeys the following recurrence relation,

Figures (1)

  • Figure :

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 3.1
  • Lemma 3.1
  • Remark 1
  • Theorem 3.1
  • proof
  • Corollary 3.1
  • Remark 2
  • Definition 4.1
  • Lemma 4.1
  • ...and 25 more