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Peer-reviewedCC BYMetrika

The distribution of the ratio of products of independent zero mean normal random variables.

  • Robert E Gaunt1
  • Heather L Sutcliffe1
  1. 1Department of Mathematics, The University of Manchester, Oxford Road, Manchester, M13 9PL UK.
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Abstract

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}X1,… ,XM\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}Y1,… ,YN\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}σ Xi2\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}σ Yj2\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=X1⋯ XM\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=Y1⋯ YN\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.

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Cite

BibTeXRIS
BibTeX
@article{gaunt2026distribution,
  title = {{The distribution of the ratio of products of independent zero mean normal random variables.}},
  author = {Robert E Gaunt and Heather L Sutcliffe},
  journal = {Metrika},
  year = {2026},
  month = aug,
  doi = {10.1007/s00184-026-01045-2},
  url = {https://europepmc.org/article/MED/42953282},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - The distribution of the ratio of products of independent zero mean normal random variables.
AU  - Robert E Gaunt
AU  - Heather L Sutcliffe
PY  - 2026
DA  - 2026/08/24
JO  - Metrika
DO  - 10.1007/s00184-026-01045-2
UR  - https://europepmc.org/article/MED/42953282
AB  - 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,… ,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,… ,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}σ _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}σ _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⋯ 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⋯ 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.
SN  - 0026-1335
ER  -