  We utilize Cauchy's argument principle in combination with the Jacobian of a
holomorphic function in several complex variables and the first moment of a
ratio of two correlated complex normal random variables to prove explicit
formulas for the density and the mean distribution of complex zeros of random
polynomials spanned by orthogonal polynomials on the unit circle and on the
unit disk. We then inquire into the consequences of their asymptotical
evaluations.
