  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-valued
on the real line, and $\eta_0,\dots,\eta_n$ are complex-valued iid Gaussian
random variables. We derive an explicit intensity function for the number of
zeros of $P_n$ in $\Omega$ for each fixed $n$. Our main application is to
polynomials orthogonal on the real line. Using the Christoffel-Darboux formula
the intensity function takes a very simple shape. Moreover, we give the
limiting value of the intensity function when the orthogonal polynomials are
associated to Szeg\H{o} weights.
