On distribution of waiting time for the first failure followed by a limited length success run

Czesław Stępniak

doi:10.4064/am40-4-3

Success runs in a two-state Markov chain

Journal of Applied Probability ◽

10.2307/3212597 ◽

1974 ◽

Vol 11 (1) ◽

pp. 190-192 ◽

Cited By ~ 20

Author(s):

M. B. Rajarshi

Keyword(s):

Markov Chain ◽

Generating Function ◽

Waiting Time ◽

Recurrent Events ◽

Bernoulli Trials ◽

Success Runs ◽

Fixed Length ◽

First Occurrence ◽

Success Run

Success runs of a fixed length in a two-state Markov chain are discussed. The results are analogous to those in the case of independent Bernoulli trials. The generating function of the waiting time for the first occurrence of a success run is obtained from the theory of recurrent events. Under certain conditions, the distribution of the number of long runs in a large number of trials is shown to be Poisson.

Download Full-text

Success runs in a two-state Markov chain

Journal of Applied Probability ◽

10.1017/s0021900200036548 ◽

1974 ◽

Vol 11 (01) ◽

pp. 190-192 ◽

Cited By ~ 4

Author(s):

M. B. Rajarshi

Keyword(s):

Markov Chain ◽

Generating Function ◽

Waiting Time ◽

Recurrent Events ◽

Bernoulli Trials ◽

Success Runs ◽

Fixed Length ◽

First Occurrence ◽

Success Run

Success runs of a fixed length in a two-state Markov chain are discussed. The results are analogous to those in the case of independent Bernoulli trials. The generating function of the waiting time for the first occurrence of a success run is obtained from the theory of recurrent events. Under certain conditions, the distribution of the number of long runs in a large number of trials is shown to be Poisson.

Download Full-text

Success Run Waiting Times and Fuss-Catalan Numbers

Journal of Probability and Statistics ◽

10.1155/2015/482462 ◽

2015 ◽

Vol 2015 ◽

pp. 1-11 ◽

Cited By ~ 1

Author(s):

S. J. Dilworth ◽

S. R. Mane

Keyword(s):

Power Series ◽

Waiting Time ◽

Geometric Distribution ◽

Waiting Times ◽

Exact Expression ◽

Catalan Numbers ◽

Auxiliary Equation ◽

Bernoulli Trials ◽

Asymptotic Results ◽

Success Run

We present power series expressions for all the roots of the auxiliary equation of the recurrence relation for the distribution of the waiting time for the first run ofkconsecutive successes in a sequence of independent Bernoulli trials, that is, the geometric distribution of orderk. We show that the series coefficients are Fuss-Catalan numbers and write the roots in terms of the generating function of the Fuss-Catalan numbers. Our main result is a new exact expression for the distribution, which is more concise than previously published formulas. Our work extends the analysis by Feller, who gave asymptotic results. We obtain quantitative improvements of the error estimates obtained by Feller.

Download Full-text