Derivation of the probability distribution functions for succession quota random variables

The probability distribution functions (pdf's) of the sooner and later waiting time random variables (rv's) for the succession quota problem (k successes and r failures) are derived presently in the case of a binary sequence of order k. The probability generating functions (pgf's) of the above rv's are then obtained directly from their pdf's. In the case of independent Bernoulli trials, expressions for the pdf's in terms of binomial coefficients are also established.     

Waiting Time Problems in a Two-State Markov Chain

Let F ₀ be the event that l ₀ 0-runs of length k ₀ occur and F ₁ be the event that l ₁ 1-runs of length k ₁ occur in a two-state Markov chain. In this paper using a combinatorial method and the Markov chain imbedding method, we obtained explicit formulas of the probability generating functions of the sooner and later waiting time between F ₀ and F ₁ by the non-overlapping, overlapping and "greater than or equal" enumeration scheme. These formulas are convenient for evaluating the distributions of the sooner and later waiting time problems.     

On the Waiting Time for the First Success Run

Let k and m are positive integers with k ≥ m. The probability generating function of the waiting time for the first occurrence of consecutive k successes in a sequence of m-th order Markov dependent trials is given as a function of the conditional probability generating functions of the waiting time for the first occurrence of consecutive m successes. This provides an efficient algorithm for obtaining the probability generating function when k is large. In particular, in the case of independent trials a simple relationship between the geometric distribution of order k and the geometric distribution of order k−1 is obtained. This research was partially supported by the ISM Cooperative Research Program(2004-ISM-CRP-2006) and by a Grant-in-Aid for Scientific Research (C) of the JSPI (Grant Number 16500183)     

Sooner and later waiting time problems for success and failure runs in higher order Markov dependent trials

The probability generating functions of the waiting times for the first success run of length k and for the sooner run and the later run between a success run of length k and a failure run of length r in the second order Markov dependent trials are derived using the probability generating function method and the combinatorial method. Further, the systems of equations of 2^.m conditional probability generating functions of the waiting times in the m-th order Markov dependent trials are given. Since the systems of equations are linear with respect to the conditional probability generating functions, they can be solved exactly, and hence the probability generating functions of the waiting time distributions are obtained. If m is large, some computer algebra systems are available to solve the linear systems of equations.This research was partially supported by the Natural Sciences and Engineering Research Council of Canada.     

On a waiting time distribution in a sequence of Bernoulli trials

In the present article we investigate the exact distribution of the waiting time for the r-th non-overlapping appearance of a pair of successes separated by at mosk k–2 failures (k2) in a sequence of independent and identically distributed (iid) Bernoulli trials. Formulae are provided for the probability distribution function, probability generating function and moments and some asymptotic results are discussed. Expressions in terms of certain generalised Fibonacci numbers and polynomials are also included.     

Waiting time problems for a sequence of discrete random variables

Let X ₁, X ₂,... be a sequence of nonnegative integer valued random variables.For each nonnegative integer i, we are given a positive integer k _i . For every i = 0, 1, 2,..., E _i denotes the event that a run of i of length k _i occurs in the sequence X ₁, X ₂,.... For the sequence X ₁, X ₂,..., the generalized pgf's of the distributions of the waiting times until the r-th occurrence among the events % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXafv3ySLgzGmvETj2BSbqefm0B1jxALjhiov2D% aebbfv3ySLgzGueE0jxyaibaiiYdd9qrFfea0dXdf9vqai-hEir8Ve% ea0de9qq-hbrpepeea0db9q8as0-LqLs-Jirpepeea0-as0Fb9pgea% 0lrP0xe9Fve9Fve9qapdbaqaaeGacaGaaiaabeqaamaabaabcaGcba% WaaiWabeaacaWGfbWaaSbaaSqaaiaadMgaaeqaaaGccaGL7bGaayzF% aaWaa0baaSqaaiaadMgacqGH9aqpcaaIWaaabaGaeyOhIukaaaaa!43D8!\[\left\{ {E_i } \right\}_{i = 0}^\infty\]are obtained. Though our situations are general, the results are very simple. For the special cases that X's are i.i.d. and {0, 1}-valued, the corresponding results are consistent with previously published results.This research was partially supported by the ISM Cooperative Research Program (90-ISM-CRP-11) of the Institute of Statistical Mathematics.     

Joint Distributions of Runs in a Sequence of Multi-State Trials

In this paper we introduce a Markov chain imbeddable vector of multinomial type and a Markov chain imbeddable variable of returnable type and discuss some of their properties. These concepts are extensions of the Markov chain imbeddable random variable of binomial type which was introduced and developed by Koutras and Alexandrou (1995, Ann. Inst. Statist. Math., 47, 743–766). By using the results, we obtain the distributions and the probability generating functions of numbers of occurrences of runs of a specified length based on four different ways of counting in a sequence of multi-state trials. Our results also yield the distribution of the waiting time problems.     

Sooner and later waiting time problems in a two-state Markov chain

LetX ₁,X ₂,... be a time-homogeneous {0, 1}-valued Markov chain. LetF ₀ be the event thatl runs of 0 of lengthr occur and letF ₁ be the event thatm runs of 1 of lengthk occur in the sequenceX ₁,X ₂, ... We obtained the recurrence relations of the probability generating functions of the distributions of the waiting time for the sooner and later occurring events betweenF ₀ andF ₁ by the non-overlapping way of counting and overlapping way of counting. We also obtained the recurrence relations of the probability generating functions of the distributions of the sooner and later waiting time by the non-overlapping way of counting of 0-runs of lengthr or more and 1-runs of lengthk or more.     

On multiparameter distributions of order k

A multiparameter negative binomial distribution of order k is obtained by compounding the extended (or multiparameter) Poisson distribution of order k by the gamma distribution. A multiparameter logarithmic series distribution of order k is derived next, as the zero truncated limit of the first distribution. Finally a few genesis schemes and interrelationships are established for these three multiparameter distributions of order k. The present work extends several properties of distributions of order k.     

Numbers of Success-Runs of Specified Length Until Certain Stopping Time Rules and Generalized Binomial Distributions of Order k

A new distribution called a generalized binomial distribution of order k is defined and some properties are investigated. A class of enumeration schemes for success-runs of a specified length including non-overlapping and overlapping enumeration schemes is rigorously studied. For each nonnegative integer less than the specified length of the runs, an enumeration scheme called -overlapping way of counting is defined. Let k and be positive integers satisfying < k. Based on independent Bernoulli trials, it is shown that the number of (– 1)-overlapping occurrences of success-run of length k until the n-th overlapping occurrence of success-run of length follows the generalized binomial distribution of order (k–). In particular, the number of non-overlapping occurrences of success-run of length k until the n-th success follows the generalized binomial distribution of order (k– 1). The distribution remains unchanged essentially even if the underlying sequence is changed from the sequence of independent Bernoulli trials to a dependent sequence such as higher order Markov dependent trials. A practical example of the generalized binomial distribution of order k is also given.