We study the distribution of complex zeros of the random harmonic polynomials with independent complex Gaussian coefficients. The expected number of zeros is evaluated following a representation for the expected absolute value of certain Gaussian quadratic forms. We then apply a similar method to find the expected number of images produced in two models of random gravitational lensing systems: the random relative mass model and the floating black hole model, based on the relations between the harmonic polynomials and the lens equations.

The techniques used in finding the expected number of zeros are later extended to determine the expectations of absolute value and sign of general Gaussian quadratic forms. Several interesting corollaries and examples are presented as well. An important structure of expected absolute value of Gaussian product is also discussed. We explore an upper bound for this structure which is related to the permanent of the covariance matrix of the random vector and we compare it with other bounds.