On the fully analytical cumulative distribution of product of correlated Gaussian random Variables with zero means
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914033106419712 |
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| author | Akyildirim, Erdinc Hekimoglu, Alper |
| author_facet | Akyildirim, Erdinc Hekimoglu, Alper |
| contents | We derive a fully analytical, one-line closed-form expression for the cumulative distribution function (CDF) of the product of two correlated zero-mean normal random variables, avoiding any series representation. This result complements the well-known compact density formula with an equally compact and computationally practical CDF representation. Our main formula expresses the CDF in terms of Humbert's confluent hypergeometric function $Φ_1$ and modified Bessel functions $K_ν$, offering both theoretical elegance and computational efficiency. High-precision numerical experiments confirm pointwise agreement with Monte Carlo simulations and other benchmarks to machine accuracy. The resulting representation provides a tractable tool for applications in wireless fading channel modeling, nonlinear signal processing, statistics, finance, and applied probability. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_09866 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the fully analytical cumulative distribution of product of correlated Gaussian random Variables with zero means Akyildirim, Erdinc Hekimoglu, Alper Probability We derive a fully analytical, one-line closed-form expression for the cumulative distribution function (CDF) of the product of two correlated zero-mean normal random variables, avoiding any series representation. This result complements the well-known compact density formula with an equally compact and computationally practical CDF representation. Our main formula expresses the CDF in terms of Humbert's confluent hypergeometric function $Φ_1$ and modified Bessel functions $K_ν$, offering both theoretical elegance and computational efficiency. High-precision numerical experiments confirm pointwise agreement with Monte Carlo simulations and other benchmarks to machine accuracy. The resulting representation provides a tractable tool for applications in wireless fading channel modeling, nonlinear signal processing, statistics, finance, and applied probability. |
| title | On the fully analytical cumulative distribution of product of correlated Gaussian random Variables with zero means |
| topic | Probability |
| url | https://arxiv.org/abs/2509.09866 |