On the fully analytical cumulative distribution of product of correlated Gaussian random Variables with zero means

2025/09/11 by Erdinç Akyıldırım, Akyildirim, Erdinc, Alper Hekimoğlu +1
Environmental Science · #Analysis of environmental and stochastic processes

paper · pdf · doi:10.48550/arxiv.2509.09866

Abstract

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 Φ₁ 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.

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