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We study zero distribution of random linear combinations of the form$$P_n(z)=\sum_{j=0}^n\eta_jf_j(z),$$ in any Jordan region $\Omega \subset\mathbb C$. The basis functions $f_j$ are entire functions that are real-valuedon the real line, and $\eta_0,\dots,\eta_n$ are complex-valued iid Gaussianrandom variables. We derive an explicit intensity function for the number ofzeros of $P_n$ in $\Omega$ for each fixed $n$. Our main application is topolynomials orthogonal on the real line. Using the Christoffel-Darboux formulathe intensity function takes a very simple shape. Moreover, we give thelimiting value of the intensity function when the orthogonal polynomials areassociated to Szeg\H{o} weights.

Zeros of random linear combinations of entire functions with complex Gaussian coefficients