{
  "abstract": "Let \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X_1,\\ldots ,X_M$$\\end{document}X1,…,XM and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y_1,\\ldots ,Y_N$$\\end{document}Y1,…,YN be independent zero mean normal random variables with variances \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _{X_i}^2$$\\end{document}σXi2, i=1,… ,Mi=1,…,M, and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\sigma _{Y_j}^2$$\\end{document}σYj2, j=1,… ,Nj=1,…,N, respectively, and let \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$X=X_1\\cdots X_M$$\\end{document}X=X1⋯XM and \\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$Y=Y_1\\cdots Y_N$$\\end{document}Y=Y1⋯YN. In this paper, we derive the exact probability density function of the ratio X/Y. We apply this formula to derive exact formulas for the cumulative distribution function and the characteristic function. We also obtain further distributional properties, including asymptotic approximations for the probability density function, tail probabilities and the quantile function.",
  "abstract_shown": true,
  "affiliations": [
    "Department of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL UK."
  ],
  "article_type": "research article",
  "arxiv": null,
  "author_total": 2,
  "authors": [
    {
      "affs": [
        0
      ],
      "name": "Robert E Gaunt",
      "orcid": "0000-0001-6187-0657"
    },
    {
      "affs": [
        0
      ],
      "name": "Heather L Sutcliffe"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5t2cd76h3q6qm.bib",
  "cited_by": 0,
  "doi": "10.1007/s00184-026-01045-2",
  "fulltext_url": "https://europepmc.org/articles/PMC13714066",
  "id": "5t2cd76h3q6qm",
  "issns": [
    "0026-1335"
  ],
  "landing_url": "https://europepmc.org/article/MED/42953282",
  "language": "eng",
  "licence": "CC-BY",
  "licence_label": "CC BY",
  "licence_source": "article",
  "licence_url": "https://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": null,
  "native_subjects": [
    [
      "heading",
      "Article"
    ],
    [
      "openalex",
      "2613|Statistical Distribution Estimation and Applications"
    ]
  ],
  "openalex": "W7125148194",
  "pdf_url": "https://europepmc.org/articles/PMC13714066?pdf=render",
  "pmcid": "PMC13714066",
  "pmid": "42953282",
  "published": "2026-08-24",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5t2cd76h3q6qm.ris",
  "sources": [
    {
      "first_seen": "2026-10-31T00:20:41Z",
      "native_id": "MED:42953282",
      "source": "europepmc",
      "url": "https://europepmc.org/article/MED/42953282"
    },
    {
      "first_seen": "2026-11-01T12:21:38Z",
      "native_id": "PMC13714066",
      "source": "pmc",
      "url": "https://doi.org/10.1007/s00184-026-01045-2"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "openalex:2613|Statistical Distribution Estimation and Applications",
      "subfield": "statistics"
    }
  ],
  "title": "The distribution of the ratio of products of independent zero mean normal random variables.",
  "title_text": "The distribution of the ratio of products of independent zero mean normal random variables.",
  "type": "peer-reviewed",
  "updated_at": "2026-11-01T12:21:38Z",
  "url": "https://research.draconitic.com/paper/5t2cd76h3q6qm",
  "venue": "Metrika",
  "versions": []
}