{
  "id": "2303.17487",
  "version": "v1",
  "published": "2023-03-30T15:57:10.000Z",
  "updated": "2023-03-30T15:57:10.000Z",
  "title": "The extreme values of two probability functions for the Gamma distribution",
  "authors": [
    "Ping Sun",
    "Ze-Chun Hu",
    "Wei Sun"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Motivated by Chv\\'{a}tal's conjecture and Tomaszewaki's conjecture, we investigate the extreme value problem of two probability functions for the Gamma distribution. Let $\\alpha,\\beta$ be arbitrary positive real numbers and $X_{\\alpha,\\beta}$ be a Gamma random variable with shape parameter $\\alpha$ and scale parameter $\\beta$. We study the extreme values of functions $P\\{X_{\\alpha,\\beta}\\le E[X_{\\alpha,\\beta}]\\}$ and $P\\{|X_{\\alpha,\\beta}-E[X_{\\alpha,\\beta}]|\\le \\sqrt{{\\rm Var}(X_{\\alpha,\\beta})}\\}$. Among other things, we show that $ \\inf_{\\alpha,\\beta}P\\{X_{\\alpha,\\beta}\\le E[X_{\\alpha,\\beta}]\\}=\\frac{1}{2}$ and $\\inf_{\\alpha,\\beta}P\\{|X_{\\alpha,\\beta}-E[X_{\\alpha,\\beta}]|\\le \\sqrt{{\\rm Var}(X_{\\alpha,\\beta})}\\}=P\\{|Z|\\le 1\\}\\approx 0.6826$, where $Z$ is a standard normal random variable.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-03-30T15:57:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60E15",
      "62G32",
      "90C15"
    ],
    "keywords": [
      "gamma distribution",
      "probability functions",
      "extreme value problem",
      "arbitrary positive real numbers",
      "standard normal random"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}