Let ${\left\{{\eta }_j\right\}}_{j=0}^N$ be a sequence of independent and identically distributed random complex Gaussian variables, and let ${\left\{f_j\left(z\right)\right\}}_{j=0}^N$ be a sequence of given analytic functions that are real-valued on the real line. We prove an exact formula for the expected density of the distribution of complex zeros of the random equation ${\sum }_{j=0}^N{\eta }_j{f}_j\left(z\right)=\mathbf{K}$, where $\mathbf{K}\in \mathbb{C}$. The method of proof employs a formula for the expected absolute value of quadratic forms of Gaussian random variables. We also obtain the limiting behaviour of the density function as $N$ tends to infinity and provide numerical computations for the density function and empirical distributions for random sums with various choices of the functions $f_j\left(z\right)$. Finally, we study the cases when the functions $f_j\left(z\right)$ are polynomials orthogonal on the real line and the unit circle.

让 ${\left\{{\eta }_{Ĵ}\right\}}_{Ĵ=0}^{ñ}$ 是一系列独立且分布均匀的随机复高斯变量，并令 ${\left\{F_{Ĵ}（ž）\right\}}_{Ĵ=0}^{ñ}$是在实线上实值的给定分析函数的序列。我们证明了随机方程复零的分布的预期密度的精确公式${\sum }_{Ĵ=0}^{ñ}{\eta }_{Ĵ}{F}_{Ĵ}（ž）=\mathbf{ķ}$，在哪里 $\mathbf{ķ}\in \mathbb{C}$。证明方法对高斯随机变量的二次形式的预期绝对值采用公式。我们还获得了密度函数的极限行为$ñ$ 趋于无穷大，并为密度函数提供了数值计算，并为函数的各种选择提供了随机和的经验分布 $F_{Ĵ}（ž）$。最后，我们研究了当功能$F_{Ĵ}（ž）$ 是在实线和单位圆上正交的多项式。