The distribution of the ratio of products of independent zero mean normal random variables
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866912832149258240 |
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| author | Gaunt, Robert E. Sutcliffe, Heather L. |
| author_facet | Gaunt, Robert E. Sutcliffe, Heather L. |
| contents | Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $σ_{X_i}^2$, $i=1,\ldots,M$, and $σ_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$. 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_12997 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The distribution of the ratio of products of independent zero mean normal random variables Gaunt, Robert E. Sutcliffe, Heather L. Probability Primary 60E05, 62E15, Secondary 33C60, 41A60 Let $X_1,\ldots,X_M$ and $Y_1,\ldots,Y_N$ be independent zero mean normal random variables with variances $σ_{X_i}^2$, $i=1,\ldots,M$, and $σ_{Y_j}^2$, $j=1,\ldots,N$, respectively, and let $X=X_1\cdots X_M$ and $Y=Y_1\cdots Y_N$. 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. |
| title | The distribution of the ratio of products of independent zero mean normal random variables |
| topic | Probability Primary 60E05, 62E15, Secondary 33C60, 41A60 |
| url | https://arxiv.org/abs/2601.12997 |