Abstract
Joint distributions of the numbers of failures, successes and success-runs of length less thank until the first consecutivek successes are obtained for some random sequences such as a sequence of independent and identically distributed integer valued random variables, a {0, 1}-valued Markov chain and a binary sequence of orderk. There are some ways of counting numbers of runs with a specified length. This paper studies the joint distributions based on three ways of counting numbers of runs, i.e., the number of overlapping runs with a specified length, the number of non-overlapping runs with a specified length and the number of runs with a specified length or more. Marginal distributions of them can be derived immediately, and most of them are surprisingly simple.
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References
Aki, S. (1985). Discrete distributions of orderk on a binary sequence,Ann. Inst. Statist. Math.,37, 205–224.
Aki, S. (1992). Waiting time problems for a sequence of discrete random variables,Ann. Inst. Statist. Math.,44, 363–378.
Aki, S. and Hirano, K. (1988). Some characteristics of the binomial distribution of orderk and related distributions,Statistical Theory and Data Analysis II, Proceedings of the 2nd Pacific Area Statistical Conference (ed. K. Matusita), 211–222, North-Holland, Amsterdam.
Aki, S. and Hirano, K. (1994). Distributions of numbers of failures and successes until the first consecutivek successes,Ann. Inst. Statist. Math.,46, 193–202.
Feller, W. (1968).An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed., Wiley, New York.
Fu, J. C. (1986). Reliability of consecutive-k-out-of-n:F systems with (k − 1)-step Markov dependence,IEEE Transactions on Reliability,R-35, 602–606.
Godbole, A. P. (1992). The exact and asymptotic distribution of overlapping success runs,Comm. Statist. Theory Methods,21(4, 953–967.
Goldstein, L. (1990). Poisson approximation and DNA sequence matching,Comm. Statist. Theory Methods,19(11), 4167–4179.
Hirano, K. and Aki, S. (1987). Properties of the extended distributions of orderk, Statist. Probab. Lett.,6, 67–69.
Hirano, K. and Aki, S. (1993). On number of occurrences of success runs of specified length in a two-state Markov chain,Statist. Sinica,3, 313–320.
Hirano, K., Aki, S., Kashiwagi, N. and Kuboki, H. (1991). On Ling's binomial and negative binomial distributions of orderk, Statist. Probab. Lett.,11, 503–509.
Johnson, N. L. and Kotz, S. (1969).Discrete Distributions, Houghton MIfflin, Boston.
Ling, K. D. (1988). On binomial distributions of orderk, Statist. Probab. Lett.,6, 247–250.
Philippou, A. N. (1988). On multiparameter distributions of orderk, Ann. Inst. Statist. Math.,40, 467–475.
Philippou, A. N., Georghiou, C. and Philippou, G. N. (1983). A generalized geometric distribution and some of its properties,Statist. Probab. Lett.,1, 171–175.
Schwager, S. J. (1983). Run probabilities in sequences of Markov-dependent trials,J. Amer. Statist. Assoc.,78, 168–175.
Xekalaki, E. and Panaretos, J. (1989). On some distributions arising in inverse cluster sampling,Comm. Statist. Theory Methods,18(1), 355–366.
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This research was partially supported by the ISM Cooperative Research Program (93-ISM-CRP-8).
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Aki, S., Hirano, K. Joint distributions of numbers of success-runs and failures until the first consecutivek successes. Ann Inst Stat Math 47, 225–235 (1995). https://doi.org/10.1007/BF00773459
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DOI: https://doi.org/10.1007/BF00773459