{
  "id": "2305.16631",
  "version": "v1",
  "published": "2023-05-26T04:54:48.000Z",
  "updated": "2023-05-26T04:54:48.000Z",
  "title": "On the maximum of the weighted binomial sum $(1+a)^{-r}\\sum_{i=0}^{r}\\binom{m}{i}a^{i}$",
  "authors": [
    "Seok Hyun Byun",
    "Svetlana Poznanović"
  ],
  "comment": "18 pages, 1 figure, 1 table. Any comments would be appreciated!",
  "categories": [
    "math.CO"
  ],
  "abstract": "Recently, Glasby and Paseman considered the following sequence of binomial sums $\\{2^{-r}\\sum_{i=0}^{r}\\binom{m}{i}\\}_{r=0} ^{m}$ and showed that this sequence is unimodal and attains its maximum value at $r=\\lfloor\\frac{m}{3}\\rfloor+1$ for $m\\in\\mathbb{Z}_{\\geq0}\\setminus\\{0,3,6,9,12\\}$. They also analyzed the asymptotic behavior of the maximum value of the sequence as $m$ approaches infinity. In the present work, we generalize their results by considering the sequence $\\{(1+a)^{-r}\\sum_{i=0}^{r}\\binom{m}{i}a^{i}\\}_{r=0} ^{m}$ for positive integers $a$. We also consider a family of discrete probability distributions that naturally arises from this sequence.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-05-26T04:54:48.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05A10"
    ],
    "keywords": [
      "weighted binomial sum",
      "maximum value",
      "discrete probability distributions",
      "approaches infinity",
      "asymptotic behavior"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}