Sooner waiting time problems in a sequence of trinary trials

1997 ◽  
Vol 34 (3) ◽  
pp. 593-609 ◽  
Author(s):  
M. V. Koutras ◽  
V. A. Alexandrou
Keyword(s):  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Generating Functions ◽  
Waiting Times ◽  
Practical Applications ◽  
Probability Generating Functions

Let X1, X2,· ·· be a (linear or circular) sequence of trials with three possible outcomes (say S, S∗ or F) in each trial. In this paper, the waiting time for the first appearance of an S-run of length k or an S∗-run of length r is systematically investigated. Exact formulae and Chen-Stein approximations are derived for the distribution of the waiting times in both linear and circular problems and their asymptotic behaviour is illustrated. Probability generating functions are also obtained when the trials are identical. Finally, practical applications of these results are discussed in some detail.

Sooner waiting time problems in a sequence of trinary trials

1997 ◽  
Vol 34 (03) ◽  
pp. 593-609 ◽  
Author(s):  
M. V. Koutras ◽  
V. A. Alexandrou
Keyword(s):  
Asymptotic Behaviour ◽  
Waiting Time ◽  
Generating Functions ◽  
Waiting Times ◽  
Practical Applications ◽  
Probability Generating Functions

Let X 1 , X 2 ,· ·· be a (linear or circular) sequence of trials with three possible outcomes (say S, S∗ or F) in each trial. In this paper, the waiting time for the first appearance of an S-run of length k or an S∗-run of length r is systematically investigated. Exact formulae and Chen-Stein approximations are derived for the distribution of the waiting times in both linear and circular problems and their asymptotic behaviour is illustrated. Probability generating functions are also obtained when the trials are identical. Finally, practical applications of these results are discussed in some detail.

scholarly journals Distributions of Patterns of Pair of Successes Separated by Failure Runs of Length at Least and at Most Involving Markov Dependent Trials: GERT Approach

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Kanwar Sen ◽  
Pooja Mohan ◽  
Manju Lata Agarwal
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Waiting Times ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Mean And Variance ◽  
Markov Dependent Trials

We use the Graphical Evaluation and Review Technique (GERT) to obtain probability generating functions of the waiting time distributions of 1st, and th nonoverlapping and overlapping occurrences of the pattern , involving homogenous Markov dependent trials. GERT besides providing visual picture of the system helps to analyze the system in a less inductive manner. Mean and variance of the waiting times of the occurrence of the patterns have also been obtained. Some earlier results existing in literature have been shown to be particular cases of these results.

On some waiting time problems

2000 ◽  
Vol 37 (03) ◽  
pp. 756-764 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Survival Analysis ◽  
Finite Number ◽  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Waiting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

On some waiting time problems

2000 ◽  
Vol 37 (3) ◽  
pp. 756-764 ◽  
Author(s):  
Valeri T. Stefanov
Keyword(s):  
Survival Analysis ◽  
Finite Number ◽  
Markov Processes ◽  
Waiting Time ◽  
Generating Functions ◽  
Continuous Time ◽  
Waiting Times ◽  
Phase Type ◽  
Phase Type Distributions ◽  
Semi Markov Processes

A unifying technology is introduced for finding explicit closed form expressions for joint moment generating functions of various random quantities associated with some waiting time problems. Sooner and later waiting times are covered for general discrete- and continuous-time models. The models are either Markov chains or semi-Markov processes with a finite number of states. Waiting times associated with generalized phase-type distributions, that are of interest in survival analysis and other areas, are also covered.

Sooner and later waiting time problems for patterns in Markov dependent trials

2003 ◽  
Vol 40 (1) ◽  
pp. 73-86 ◽  
Author(s):  
Qing Han ◽  
Katuomi Hirano
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Random Variables ◽  
Probability Functions ◽  
Probability Generating Functions ◽  
Markov Dependent Trials

In this paper, we investigate sooner and later waiting time problems for patterns S0 and S1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of S0 and between successive occurrences of S0 and S1 and of the waiting time until the rth occurrence of S0 are also given.

Sooner and later waiting time problems for patterns in Markov dependent trials

2003 ◽  
Vol 40 (01) ◽  
pp. 73-86 ◽  
Author(s):  
Qing Han ◽  
Katuomi Hirano
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Random Variables ◽  
Probability Functions ◽  
Probability Generating Functions ◽  
Markov Dependent Trials

In this paper, we investigate sooner and later waiting time problems for patterns S 0 and S 1 in multistate Markov dependent trials. The probability functions and the probability generating functions of the sooner and later waiting time random variables are studied. Further, the probability generating functions of the distributions of distances between successive occurrences of S 0 and between successive occurrences of S 0 and S 1 and of the waiting time until the rth occurrence of S 0 are also given.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

2002 ◽  
Vol 39 (1) ◽  
pp. 70-80 ◽  
Author(s):  
James C. Fu ◽  
Y. M. Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

scholarly journals On Generating Functions of Waiting Times and Numbers of Occurrences of Compound Patterns in a Sequence of Multistate Trials

2007 ◽  
Vol 44 (01) ◽  
pp. 71-81 ◽  
Author(s):  
Kiyoshi Inoue ◽  
Sigeo Aki
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Waiting Times ◽  
Fixed Number ◽  
Statistics And Probability ◽  
Time Problem ◽  
Waiting Time Problem ◽  
Theoretical Results

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

On probability generating functions for waiting time distributions of compound patterns in a sequence of multistate trials

2002 ◽  
Vol 39 (01) ◽  
pp. 70-80 ◽  
Author(s):  
James C. Fu ◽  
Y. M. Chang
Keyword(s):  
Markov Chain ◽  
Waiting Time ◽  
Generating Functions ◽  
Probability Generating Function ◽  
Finite Markov Chain Imbedding ◽  
Waiting Time Distributions ◽  
Probability Generating Functions ◽  
Markov Chain Imbedding ◽  
Markov Chain Imbedding Method ◽  
Runs And Patterns

Probability generation functions of waiting time distributions of runs and patterns have been used successfully in various areas of statistics and applied probability. In this paper, we provide a simple way to obtain the probability generating functions for waiting time distributions of compound patterns by using the finite Markov chain imbedding method. We also study the characters of waiting time distributions for compound patterns. A computer algorithm based on Markov chain imbedding technique has been developed for automatically computing the distribution, probability generating function, and mean of waiting time for a compound pattern.

scholarly journals On Generating Functions of Waiting Times and Numbers of Occurrences of Compound Patterns in a Sequence of Multistate Trials

2007 ◽  
Vol 44 (1) ◽  
pp. 71-81 ◽  
Author(s):  
Kiyoshi Inoue ◽  
Sigeo Aki
Keyword(s):  
Waiting Time ◽  
Generating Functions ◽  
Waiting Times ◽  
Fixed Number ◽  
Statistics And Probability ◽  
Time Problem ◽  
Waiting Time Problem ◽  
Theoretical Results

In this paper we study two distributions, namely the distribution of the waiting times until given numbers of occurrences of compound patterns and the distribution of the numbers of occurrences of compound patterns in a fixed number of trials. We elucidate the interrelation between these two distributions in terms of the generating functions. We provide perspectives on the problems related to compound patterns in statistics and probability. As an application, the waiting time problem of counting runs of specified lengths is considered in order to illustrate how the distributions of waiting times can be derived from our theoretical results.

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