Type IIIIIIandIV q-negative binomial distributions of order k Jungtaek Oh

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Type I,II,III and IV q-negative binomial

distributions of order k

Jungtaek Oh*

March 19, 2024

Abstract

The distributions of waiting times in variations of the negative binomial distribution of

order k. In one variation, we apply different enumeration schemes on the runs of successes.

In another variation, binary trials with a geometrically varying probability of ones were per-

formed. The exact distribution of the waiting time for the r-th occurrence of a success run of

a speciﬁed length (e.g., nonoverlapping, overlapping, at least, exact, and ℓ-overlapping) in a

q-sequence of binary trials. First, the waiting time for the r-th occurrence of a success run

with the ”nonoverlapping” counting scheme was examined. Theorem 3.1 gives a probability

function of the Type I q-negative binomial distribution of order k. Next, we consider the

waiting time for the r-th occurrence of a success run with the ”at least” counting scheme.

Theorem 4.1 gives a probability function of the Type II q-negative binomial distribution of

order k. Next, we consider the waiting time for the r-th occurrence of a success run with the

”overlapping” counting scheme. Theorem 5.1 gives a probability function of the Type III

q-negative binomial distribution of order k. Next, we consider the waiting time for the r-th

occurrence of a success run with the ”exact” counting scheme. Theorem 6.1 gives a proba-

bility function of the Type IV q-negative binomial distribution of order k. Next, we consider

the waiting time for the r-th occurrence of a success run with the ”ℓ-overlapping” counting

scheme, which is a more generalized counting scheme. Theorem 7.1 gives a probability

function of the q-negative binomial distribution of order kin the ℓ-overlapping case.

The main theorems are Type I,II,III, and IV q-negative binomial distributions of order

kand the q-negative binomial distribution of order kin the ℓ-overlapping case. This work,

examined sequences of independent binary zero and one trials with not necessarily identical

distribution with the probability of ones varying according to a geometric rule. The exact

formulas for the distributions were obtained using enumerative combinatorics.

Keywords: Waiting time problems, Type I,II,III, and IV q-negative binomial distribution of

order k,q-Negative binomial distribution of order kin the ℓ-overlapping case, Runs, Binary

trials, q-Distributions

*Corresponding Author: Department of Biomedical Science, Kyungpook National University, Daegue, 41566,

Republic of Korea (e-mail: jungtaekoh0191@gmail.com)

arXiv:2210.03617v4 [math.PR] 18 Mar 2024

1 Introduction

Charalambides (2010b) studied discrete q-distributions on Bernoulli trials with geometrically

varying success probabilities. Let us consider a sequence X1,...,Xnof zero(failure)-one(success)

Bernoulli trials, such that the trials of the subsequence after the (i−1)-th zero until the i-th

zero are independent with equal failure probability. The i-th geometric sequences of trials is the

subsequence after the (i−1)-th zero and until the i-th zero, for i>0, and the subsequence after

the (j−1)’st zero and until the j’th zero, for j>0 are independent for all i̸=j(i.e., the i-th

and j-th geometric sequences are independent) with probability of zeroes at the i-th geometric

sequence of trials.

qi=1−θqi−1,i=1,2,..., 0≤θ≤1,0≤q<1.(1.1)

The probability of failures in the independent geometric sequences of trials increases geometri-

cally with rate q. Let Fj=Σj

m=1(1−Xm)denote the number of zeroes in the ﬁrst jtrials. Because

the probability of a zero at the i-th geometric sequence of trials is in fact the conditional probabil-

ity of the occurrence of a zero at any trial jgiven the occurrence of (i−1)zeroes in the previous

trials. We can express this situation as follows.

qj,i=pXj=0Fj=i−1=1−θqi−1,i=1,2,..., j,j=1,2,.... (1.2)

Here, (1.1) is exactly equal to the conditional probability given in (1.2). For a clear understand-

ing, we consider an example with n=18, i.e., the binary sequence 111011110111100110. Each

subsequence has its own success and failure probabilities according to a geometric rule.

[1110][11110][11110][0][110]

𝜃𝑞1-𝜃𝑞𝜃𝑞1-𝜃𝑞𝜃𝑞1-𝜃𝑞1-𝜃𝑞𝜃𝑞1-𝜃𝑞

The stochastic model (1.1) or (1.2) has interesting applications. It was studied as a reliability

growth model by Dubman and Sherman (1969) and applied to a q-boson theory in physics by

Jing and Fan (1994) and Jing (1994). More speciﬁcally, the q-binomial distribution was intro-

duced as a q-deformed binomial distribution, to set up a q-binomial state. The stochastic model

(1.1) was also applied as a sequential-intervention model to start-up demonstration tests, as pro-

posed by Balakrishnan et al. (1995).

The stochastic model (1.1) is the q-analog of the classical binomial distribution with geomet-

rically varying probability of zeroes, which is a stochastic model of independent and identically

distributed (IID) trials with the following failure probability:

πj=PXj=0=1−θ,j=1,2,..., 0<θ<1.(1.3)

As qtends toward 1, the stochastic model (1.1) reduces to IID (Bernoulli) model (1.3), because

qi→πi,i=1,2,... or qj,i→1−θ,i=1,2,..., j,j=1,2,....

Discrete q-distributions based on the stochastic model of the sequence of independent Bernoulli

trials have been explored by numerous researchers. For a lucid review and comprehensive list

of publications in this area, the interested reader may consult the monographs by Charalambides

(2010b,a, 2016).

From mathematical and statistical viewpoints, Charalambides (2016), in the preface of his

book remarked upon the advantage of using a stochastic model of a sequence of independent

Bernoulli trials, in which the probability of success in a trial is assumed to vary with the number

of trials and/or the number of successes. Such a model is advantageous because it allows us to

incorporate the experience gained from previous trials and/or successes. If the probability of suc-

cess at a trial is a general function of the number of trials and/or the number of successes, little

can be inferred about the distributions of the various random variables that cam possiblt be de-

ﬁned using this model. The assumption that the probability of success (or failure) at a trial varies

geometrically, with the rate (proportion) q, leads to the introduction of discrete q-distributions.

Let us consider a distribution that is related to the success run analog of the classical negative

binomial distribution. Let X1,X2,... be a sequence of binary trials with two possible outcomes

(i.e., success or failure) in each trial. In this work, we examined the waiting time until the r-th

(where ris a positive integer) appearance of a success run of length kby considering the enu-

meration scheme (i.e., nonoverlapping, at least, overlapping, exact, or ℓ-overlapping scheme).

Notably, the special case r=1 reduces to the geometric distribution of order k(the distribution

of the number of trials until the success run of length k, denoted as Tk). When considering the

waiting distribution, different counting schemes are used, and each scheme generates a different

type of waiting time distribution.

There are several ways to count a scheme. Each counting scheme depends on different condi-

tions: whether overlapping counting is permitted and whether counting starts from scratch when

a certain kind or size of run has been enumerated. Feller (1968) proposed a classical counting

method: once kconsecutive successes are observed, the number of occurrences of kconsecutive

successes is counted, and the counting procedure starts anew (i.e., from scratch). This is called a

nonoverlapping counting scheme or Type Idistributions of order k. In another scheme, success

runs of length greater than or equal to kpreceded and followed by a failure or by the beginning

or by the end of the sequence are counted(see., e.g., Mood (1940)), and this is usually called

the at least counting scheme or Type II distribution of order k. Ling (1988) suggested the over-

lapping counting scheme, where in an uninterrupted sequence of m≥ksuccesses preceded and

followed by a failure or by the beginning or by the end of the sequence are counted. It accounts

for m−k+1 success runs of length of kand is also referred to as Type III distributions of order

k. Mood (1940) suggested an exact counting scheme, wherein success runs of length exactly k

preceded and succeeded by failure or by nothing are counted. This scheme is also referred to as

Type IV distributions of order k.

It is well known that the negative binomial distribution arises as the distribution of the sum of r

independent random variables following the geometric distribution with parameter p. The ran-

dom variable W(a)

r,kdenoted by the waiting time for the r-th occurrence of a success run with the

counting scheme used a=I, which indicates the nonoverlapping counting scheme; a=II, which

indicates the at least counting scheme; a=III, which indicates the overlapping one; and a=IV ,

which indicates the exactly one these schemes are denoted as W(I)

r,k,W(II)

r,k,W(III)

r,k, and W(IV )

r,k, re-

spectively. In addition, if the sequence is an IID sequence of random variables X1,X2,..., then the

distributions of W(I)

r,k,W(II)

r,k,W(III)

r,kand W(IV )

r,kwill be referred to as Type I,II,III and IV negative

binomial distributions of order kand will be denoted as NB(I)

k(r,θ),NB(II)

k(r,θ),NB(III)

k(r,θ),

and NB(IV )

k(r,θ), respectively.

When the sequence is a q-geometric model, the distributions of W(I)

r,k,W(II)

r,k,W(III)

r,kand W(IV )

r,k

have been called Type I,II,III and IV q-negative binomial distributions of order k, respectively.

They can be denoted as q−NB(I)

k(r,θ),q−NB(II)

k(r,θ),q−NB(III)

k(r,θ)and q−NB(IV )

k(r,θ),

respectively.

According to the four aforementioned counting schemes, the random variables of the num-

ber of runs of length kcounted in noutcomes have four distributions denoted as Nn,k,Gn,k,

Mn,k, and En,k, respectively. Moreover, if the sequence is an IID sequence of random variables,

X1,X2,...,Xn, then the distributions of Nn,k,Gn,k,Mn,k, and En,kwill be referred to as Type I,

II,III, and IV binomial distributions of order kand will be denoted as B(I)

k(n,θ),B(II)

k(n,θ),

B(III)

k(n,θ), and B(IV )

k(n,θ), respectively.

When the sequence is a q-geometric model, the distributions of Nn,k,Gn,k,Mn,k, and En,kare

called Type I,II,III, and IV q-binomial distribution of order k, respectively. These distributions

can be denoted as q−B(I)

k(n,θ),q−B(II)

k(n,θ),q−B(III)

k(n,θ), and q−B(IV )

k(n,θ), respectively.

To further clarify the distinctions among the aforementioned counting methods, we consider

the example n=12; in this case, the binary sequence 011111000111 contains N12,2=3, G12,2=

2, M12,2=6, E12,5=1, W(I)

2,2=5, W(II)

2,2=11, W(III)

2,2=4, and W(IV )

2,3>12.

Aki and Hirano (2000) introduced a more generalized counting scheme called the ℓ-overlapping

counting scheme, where ℓis a nonnegative integer less than k(see also Han and Aki (2000);

Antzoulakos (2003) ; Inoue and Aki (2003); Makri and Philippou (2005); Makri et al. (2007a)

; Makri and Psillakis (2015)). This scheme counts a success run of length keach of which may

have an overlapping (common) part of length at most ℓ(ℓ=0,1,...,k−1) with the previous run

of success of length kthat has already been enumerated. The nonoverlapping case (ℓ=0) and

the overlapping case (ℓ=k−1) are special cases of this scheme.

The random variable Wr,k,ℓ denotes the waiting time for the r-th occurrence of the ℓ-overlapping

success run of length k. If the sequence is an IID sequence of random variables Wr,k,ℓ, it will be

referred to as a negative binomial distribution of order kin the ℓ-overlapping case and denoted

as NBk,ℓ(r,θ).

According to the counting scheme mentioned earlier, the random variables of the number of ℓ-

overlapping success runs of length kcounted in noutcomes are denoted as Nn,k,ℓ. Moreover, if

the sequence is an IID sequence of random variables, X1,X2,...,Xn, then distributions of Nn,k,ℓ

will be referred to as binomial distributions of order kin the ℓ-overlapping case and denoted as

Bk,ℓ(n,θ).

When the sequence is a q-geometric model, the distributions of Wr,k,ℓ and Nn,k,ℓ are called q-

negative binomial distribution of order kin the ℓ-overlapping case and q-binomial distribution

of order kin the ℓ-overlapping case, respectively. They are denoted as q−NBk,ℓ(r,θ)and

q−Bn,k,ℓ(n,θ), respectively.

To explain this better, let us assume that n = 15 binary trials, numbered from 1 to 15, are per-

formed, and we obtain the following outcomes 111111011110111. Then, the ℓ-overlapping 1-

runs of length 4 are as follows: 1,2,3,4; 3,4,5,6; 8,9,10,11 for ℓ=2, and 1,2,3,4; 2,3,4,5; 3,4,5,6;

8,9,10,11 for ℓ=3. Hence, N15,4,2=3 and N15,4,3=4.

Let N(a)

n,a=I,II,III be a random variable denoting the number of occurrences of runs in the

sequence of ntrials; N(a)

n,a=I,II,III, which is coincident with Nn,k,Gn,k, and Mn,k, respectively.

The random variable Nnis closely related to the random variable Wr,k(see Feller (1968)). We

have the following dual relationship N(a)

n<rif and only if W(a)

r,k>nfor a=I,II,III. Now, the

probability function of the q-binomial distribution of order k, can be derived easily using the dual

relationship between the binomial and negative binomial distributions of order k:

q,θN(I)

n<r=P

q,θW(I)

r,k>n,P

q,θN(II)

n<r=P

q,θW(II)

r,k>n

q,θN(III)

n<r=P

q,θW(III)

r,k>n.

However, the above mentioned dual relationship cannot be considered by the Type IV enumera-

tion scheme. Because W(IV )

r,k>nimplies N(IV )

n<r, instead of an iff relationship.

The q-negative binomial distribution with parameters q,θ,k, and ris a distribution of the

length of a sequence of Bernoulli trials with geometrically increasing failure probability until the

r-th appearance of a success run of length k. Here, 0 <q,θ<1, and rand kare positive integers.

In this study, we examined the waiting time distribution for the r-th appeareance of a success

run of length k(nonoverlapping, at least, overlapping, and ℓ-overlapping), and the probability of

ones vary according to a geometric rule. The remainder of this paper is organized as follows: In

Section 2, we introduce the basic deﬁnitions and necessary notations that will be useful through-

out this article. In Section 3, we will study the Type I q-negative binomial distribution of order

k. We derive the exact probability function of the Type I q-negative binomial distribution of

order kvia combinatorial analysis. In Section 4, we will study the Type II q-negative binomial

distribution of order k. We derived the exact probability function of the Type II q-negative bi-

nomial distribution of order kvia combinatorial analysis. In Section 5, we examine the Type III

q-negative binomial distribution of order k. We derive the exact probability function of the Type

III q-negative binomial distribution of order kvia combinatorial analysis. In Section 6, we will

study the Type IV q-negative binomial distribution of order k. We derive the exact probability

function of the Type IV q-negative binomial distribution of order kusing combinatorial analysis.

In Section 7, we examine the waiting time for the r-th ℓ-overlapping occurrence of a success run

of length kin a q-geometric sequence. The exact probability function of the q-negative binomial

distribution of order kin the ℓ-overlapping case was derived via combinatorial analysis.

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TypeI,II,IIIandIVq-negativebinomialdistributionsoforderkJungtaekOh*March19,2024AbstractThedistributionsofwaitingtimesinvariationsofthenegativebinomialdistributionoforderk.Inonevariation,weapplydifferentenumerationschemesontherunsofsuccesses.Inanothervariation,binarytrialswithageometricallyvaryingpro...

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Type IIIIIIandIV q-negative binomial distributions of order k Jungtaek Oh

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