IDEAS home Printed from https://ideas.repec.org/a/eee/stapro/v194y2023ics0167715222002577.html
   My bibliography  Save this article

On the Chvátal–Janson conjecture

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0167715222002577
    Download Restriction: Full text for ScienceDirect subscribers only

    File URL: https://libkey.io/10.1016/j.spl.2022.109744?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    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).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    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).
    2. Idir Arab & Paulo Eduardo Oliveira & Tilo Wiklund, 2021. "Convex transform order of Beta distributions with some consequences," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 75(3), pages 238-256, August.
    3. Janson, Svante, 2021. "On the probability that a binomial variable is at most its expectation," Statistics & Probability Letters, Elsevier, vol. 171(C).
    4. Kapelko, Rafał, 2022. "On the moment absolute deviation of order statistics from uniform distribution," Statistics & Probability Letters, Elsevier, vol. 181(C).
    5. 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.
    6. Narayanaswamy Balakrishnan & Efe A. Ok & Pietro Ortoleva, 2021. "Inferential Choice Theory," Working Papers 2021-60, Princeton University. Economics Department..
    7. Pinelis, Iosif, 2021. "Best lower bound on the probability of a binomial exceeding its expectation," Statistics & Probability Letters, Elsevier, vol. 179(C).
    8. 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.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:stapro:v:194:y:2023:i:c:s0167715222002577. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.