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A study on the Poisson, geometric and Pascal distributions motivated by Chvátal’s conjecture

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  • Li, Fu-Bo
  • Xu, Kun
  • Hu, Ze-Chun

Abstract

Let B(n,p) denote a binomial random variable with parameters n and p. Vašek Chvátal conjectured that for any fixed n≥2, as m ranges over {0,…,n}, the probability qm≔P(B(n,m/n)≤m) is the smallest when m is closest to 2n3. This conjecture has been solved recently. Motivated by this conjecture, in this paper, we consider the corresponding minimum value problem on the probability that a random variable is not more than its expectation, when its distribution is the Poisson distribution, the geometric distribution or the Pascal distribution.

Suggested Citation

  • Li, Fu-Bo & Xu, Kun & Hu, Ze-Chun, 2023. "A study on the Poisson, geometric and Pascal distributions motivated by Chvátal’s conjecture," Statistics & Probability Letters, Elsevier, vol. 200(C).
    • Handle: RePEc:eee:stapro:v:200:y:2023:i:c:s0167715223000950
    DOI: 10.1016/j.spl.2023.109871
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    References listed on IDEAS

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    1. Doerr, Benjamin, 2018. "An elementary analysis of the probability that a binomial random variable exceeds its expectation," Statistics & Probability Letters, Elsevier, vol. 139(C), pages 67-74.
    2. Pelekis, Christos & Ramon, Jan, 2016. "A lower bound on the probability that a binomial random variable is exceeding its mean," Statistics & Probability Letters, Elsevier, vol. 119(C), pages 305-309.
    3. Barabesi, Lucio & Pratelli, Luca & Rigo, Pietro, 2023. "On the Chvátal–Janson conjecture," Statistics & Probability Letters, Elsevier, vol. 194(C).
    4. Greenberg, Spencer & Mohri, Mehryar, 2014. "Tight lower bound on the probability of a binomial exceeding its expectation," Statistics & Probability Letters, Elsevier, vol. 86(C), pages 91-98.
    5. Janson, Svante, 2021. "On the probability that a binomial variable is at most its expectation," Statistics & Probability Letters, Elsevier, vol. 171(C).
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