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International Encyclopedia of Statistical Science pp 400–402Cite as

Distributions of Order k

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The distributions of order k are infinite families of probability distributions indexed by a positive integer k, which reduce to the respective classical probability distributions for k = 1, and they have many applications. We presently discuss briefly the geometric, negative binomial, Poisson, logarithmic series and binomial distributions of order k.

Geometric Distribution of Order k

Denote by T k the number of independent Bernoulli trials with success (S) and failure (F) probabilities p and q = 1 − p (0 < p < 1), respectively, until the occurrence of the kth consecutive success. Philippou and Muwafi (1982) observed that a typical element of the event T k = x is an arrangement

$${a}_{1}{a}_{2}\ldots {a}_{{x}_{1}+{x}_{2}+\cdots +{x}_{k}}\underbrace{ SS\ldots S}_{k}\,,$$
(1)

such that x 1 of the a’s are E 1 = F, x 2 of the a’s are E 2 = SF, …, x k of the a’s are \(E_k = \underbrace{SS \ldots S}_{k-1}F\), and proceeded to obtain the following exact formula for the probability mass...

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References and Further Reading

  • Aki S, Kuboki H, Hirano K (1984) On discrete distributions of order k. Ann Inst Stat Math 36:431–440

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  • Antzoulakos DL, Chadjiconstantinidis S (2001) Distributions of numbers of success runs of fixed length in Markov dependent trials. Ann Inst Stat Math 53:599–619

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  • Antzoulakos DL, Philippou AN (1999) Multivariate Pascal polynomials of order k with probability applications. In: Howard FT (ed) Applications of Fibonacci numbers, vol 8. Kluwer, Dordrecht, pp 27–41

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  • Balakrishnan N, Koutras MV (2002) Runs and scans with applications. Wiley, New York

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  • Feller W (1968) An introduction to probability theory and its applications, vol I, 3rd edn. Wiley, New York

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  • Godbole AP (1990) Specific formulae for some success run distributions. Stat Probab Lett 10:119–124

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  • Hirano K (1986) Some properties of the distributions of order k. In: Philippou AN et al (eds) Fibonacci numbers and their applications. Reidel, Dordrecht, pp 43–53

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  • Philippou AN (1984) The negative binomial distribution of order k and some of its properties. Biom J 26:784–789

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  • Philippou AN, Georghiou C (1989) Convolutions of Fibonacci-type polynomials of order k and the negative binomial distributions of order k. Fibonacci Q 27:209–216

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  • Philippou AN, Makri FS (1985) Longest success runs and Fibonacci-type polynomials. Fibonacci Q 23:338–346

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  • Philippou AN, Makri FS (1986) Successes, runs and longest runs. Stat Probab Lett 4:211–215

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  • Philippou AN, Muwafi AA (1982) Waiting for the kth consecutive success and the Fibonacci sequence of order k. Fibonacci Q 20:28–32

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  • Philippou AN, Georghiou C, Philippou GN (1983) A generalized geometric distribution and some of its properties. Stat Probab Lett 1:171–175

    MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Patras, Patras, Greece

    Andreas N. Philippou (Professor of Probability and Statistics) & Demetrios L. Antzoulakos (Associate Professor)

Authors
  1. Andreas N. Philippou
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  2. Demetrios L. Antzoulakos
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Editor information

Editors and Affiliations

  1. Department of Statistics and Informatics, Faculty of Economics, University of Kragujevac, City of Kragujevac, Serbia

    Miodrag Lovric

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Philippou, A.N., Antzoulakos, D.L. (2011). Distributions of Order k . In: Lovric, M. (eds) International Encyclopedia of Statistical Science. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04898-2_215

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