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On the Chvátal–Janson conjecture

Author

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  • Barabesi, Lucio
  • Pratelli, Luca
  • Rigo, Pietro

Abstract

Let qm=P(X≤m), where m is a positive integer and X a binomial random variable with parameters n and m/n. Vašek Chvátal conjectured that, for fixed n≥2, qm attains its minimum when m is the integer closest to 2n/3. As shown by Svante Janson, this conjecture is true for large n. Here, we prove that the conjecture is actually true for every n≥2.

Suggested Citation

  • Barabesi, Lucio & Pratelli, Luca & Rigo, Pietro, 2023. "On the Chvátal–Janson conjecture," Statistics & Probability Letters, Elsevier, vol. 194(C).
    • Handle: RePEc:eee:stapro:v:194:y:2023:i:c:s0167715222002577
    DOI: 10.1016/j.spl.2022.109744
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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. 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.
    4. 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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    Cited by:

    1. 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).

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