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  • 1 Department of Statistics University of Delhi Delhi, 110007, India
  • 2 Department of Statistics University of Delhi Delhi, 110007, India
  • 3 Department of Statistics University of Delhi Delhi, 110007, India
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In this paper, joint distributions of number of success runs of length k and number of failure runs of length k' are obtained by using combinatorial techniques including lattice path approach under Pólya-Eggenberger model. Some of its particular cases, for different values of the parameters, are derived. Sooner and later waiting time problems and joint distributions of number of success runs of various types until first occurrence of consecutive success runs of specified length under the model are obtained. The sooner and later waiting time problems for Bernoulli trials (see Ebneshahrashoob and Sobel [3]) and joint distributions of the type discussed by Uchiada and Aki [11] are shown as particular cases. Assuming Ln and Sn to be the lengths of longest and smallest success runs, respectively, in a sample of size n drawn by Pólya-Eggenberger sampling scheme, the joint distributions of Ln and  Sn as well as distribution of M=max(Ln,Fn)n, where Fn is the length of longest failure run, are also  obtained.

  • SEN, K., AGARWAL, M. and CHAKRABORTY, S., Lattice path approach to generalized Pólya-Eggenberger model of order k, J. Statist. Planning Inference 102 (2002), 467-476.

    'Lattice path approach to generalized Pólya-Eggenberger model of order ' () 102 Planning Inference : 467 -476.

    • Search Google Scholar
  • SEN, K. and GOYAL, B., Distributions of the numbers of success runs of length k and failure runs of length k 1 J. Indian Statist. Assoc. 38 (2000), 1-22.

    'Distributions of the numbers of success runs of length k and failure runs of length k1 ' () 38 J. Indian Statist. Assoc. : 1 -22.

    • Search Google Scholar
  • LING, K. D., On binomial distributions of order k, Statist. Probab. Lett. 6 (1988), 247-250. MR 89e:60022

    'On binomial distributions of order ' () 6 Probab. Lett. : 247 -250.

  • FELLER, W., An introduction to probability theory and its applications, Vol. 1, 3rd edition, John Wiley & Sons, Inc., New York-London-Sydney, 1968. MR 37 # 3604

    An introduction to probability theory and its applications , ().

  • GIBBONS, J.D., Nonparametric statistical inference, Second edition, Statistics: Textbooks and Monographs, 65, Marcel Dekker, Inc., New York, 1985. MR 86m:62067

    Nonparametric statistical inference , ().

  • GOLDSTEIN, L., Poisson approximation and DNA sequence matching, Comm. Statist. Theory Methods 19 (1990), 4167-4179.

    'Poisson approximation and DNA sequence matching ' () 19 Comm. Statist. Theory Methods : 4167 -4179.

    • Search Google Scholar
  • SCHUSTER, E. F., Exchangeability and recursion in the conditional distribution theory of numbers and length of runs, Runs and patterns in probability: selected papers, Math. Appl. 283, Kluwer Acad. Publ., Dordrecht, 1994, 91-118. MR 95m:60018

    'Exchangeability and recursion in the conditional distribution theory of numbers and length of runs ' () 283 Runs and patterns in probability: selected papers, Math. Appl. : 91 -118.

    • Search Google Scholar
  • UCHIADA, M. and AKI, S., Sooner and later waiting time problems in a two-state Markov chain, Ann. Inst. Statist. Math. 47 (1995), 415-433. MR 97c:60173

    'Sooner and later waiting time problems in a two-state Markov chain ' () 47 Ann. Inst. Statist. Math. : 415 -433.

    • Search Google Scholar
  • RIORDAN, J., An introduction to combinatorial analysis, Wiley Publications in Mathematical Statistics, John Wiley & Sons, Inc., New York; Chapman & Hall, Ltd., London, 1958. MR 20 #3077. Reprinted: Princeton University Press, Princeton, NJ, 1980. MR 81e:05002

    An introduction to combinatorial analysis , ().

  • AKI, S. and HIRANO, K., Discrete distributions related to the succession events in a two-state Markov chain, Statistical sciences and data analysis (Tokyo, 1991), VSP International Science Publishers, Utrecht, 1993, 467-474. MR 96b:62022

    Discrete distributions related to the succession events in a two-state Markov chain, Statistical sciences and data analysis (Tokyo, 1991) , () 467 -474.

    • Search Google Scholar
  • AKI, S. and HIRANO, K'., Joint distributions of numbers of success-runs and failures until the first consecutive k-successes, Ann. Inst. Statist. Math. 47 (1995), 225-235. MR 96k:62031

    'Joint distributions of numbers of success-runs and failures until the first consecutive k-successes ' () 47 Ann. Inst. Statist. Math. : 225 -235.

    • Search Google Scholar
  • EBNESHAHRASHOOB, M. and SOBEL, M., Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas, Statist. Probab. Lett. 9 (1990), 5-11. MR 91e:60027

    'Sooner and later waiting time problems for Bernoulli trials: frequency and run quotas ' () 9 Statist. Probab. Lett. : 5 -11.

    • Search Google Scholar