{
  "annotation_id": "93be3794-6e7d-4873-b16c-4dc21ad58f0b",
  "date_created": "2026-02-17T05:53:20.460000Z",
  "date_modified": "2026-02-17T05:53:20.460000Z",
  "file_hash": "f388c1c11192ca372339e829842e23c9c5eedcce051db2b6a4a9d2dbb5cffa10",
  "private": false,
  "record": {
    "abstract": "Let $n,k$ be positive integers such that $n\\geq k$, and let $H$ be a hypergeometric random variable counting the number of black marbles in a sample without replacement of size $k$ from an urn that contains $i\\in \\{1,\\ldots, n\\}$ black and $n - i$ white marbles. It is shown that \\[ \\mathbb{P}(H \\ge \\mathbb{E}(H)) \\ge k/n\\, , \\, \\text{when} \\,\\, n\\ge 8k \\, . \\] Furthermore, provided that $1\\le \\mathbb{E}(H)\\le \\min\\{i,k\\}-2$ as well as that $\\frac{(n-i)(n-k)}{n}\u003e1$, it is shown that \\[ \\mathbb{P}(H\\ge \\mathbb{E}(H)) \\,\\ge\\, \\frac{e^{-1/8}}{4\\sqrt{2}} \\cdot \\sqrt{\\frac{n-1}{n}} \\cdot\\frac{ \\sqrt{\\text{Var}(H)} }{1 + \\sqrt{1+ \\frac{n-1}{n-k}\\cdot\\text{Var}(H)}}\\, . \\] Auxiliary results which may be of independent interest include an upper bound on the tail conditional expectation and a lower bound on the mean absolute deviation of the hypergeometric distribution.",
    "arxiv_id": "2601.09485",
    "authors": [
      "Jianhang Ai",
      "Christos Pelekis"
    ],
    "categories": [
      "math.PR"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "On lower bounds for hypergeometric tails",
    "url": "https://arxiv.org/abs/2601.09485",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "4d8cf79f-67ec-45f7-ae46-34a14bcc6714",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}