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On a Parametrization of Partial-Sums Discrete Probability Distributions

Author

Listed:
  • Ján Mačutek

    (Mathematical Institute, Slovak Academy of Sciences, Department of Mathematics, Constantine the Philosopher University in Nitra, 949 01 Nitra, Slovakia
    These authors contributed equally to this work.)

  • Gejza Wimmer

    (Mathematical Institute, Slovak Academy of Sciences, 814 73 Bratislava, Slovakia
    These authors contributed equally to this work.)

  • Michaela Koščová

    (Mathematical Institute, Slovak Academy of Sciences, 814 73 Bratislava, Slovakia
    These authors contributed equally to this work.)

Abstract

For every discrete probability distribution, there is one and only one partial summation which leaves the distribution unchanged. This invariance property is reconsidered for distributions with one parameter. We show that if we change the parameter value in the function which defines the summation, two families of distributions can be observed. The first of them consists of distributions which are resistant to the change. For these distributions, the change of the parameter is reversed by the normalization constant, and the distributions remain unchanged. The other contains distributions sensitive to the change. Partial summations with the changed parameter value applied to sensitive distributions result in new distributions with two parameters. A necessary and sufficient condition for a distribution to be resistant to the parameter change is presented.

Suggested Citation

  • Ján Mačutek & Gejza Wimmer & Michaela Koščová, 2022. "On a Parametrization of Partial-Sums Discrete Probability Distributions," Mathematics, MDPI, vol. 10(14), pages 1-8, July.
    • Handle: RePEc:gam:jmathe:v:10:y:2022:i:14:p:2476-:d:864196
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    References listed on IDEAS

    as
    1. Willmot, Gord, 1986. "Mixed Compound Poisson Distributions," ASTIN Bulletin, Cambridge University Press, vol. 16(S1), pages 59-79, April.
    2. Lin, X. Sheldon & Willmot, Gordon E., 1999. "Analysis of a defective renewal equation arising in ruin theory," Insurance: Mathematics and Economics, Elsevier, vol. 25(1), pages 63-84, September.
    3. Unnikrishnan Nair, N. & Hitha, N., 1989. "Characterization of discrete models by distribution based on their partial sums," Statistics & Probability Letters, Elsevier, vol. 8(4), pages 335-337, September.
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