  We study the expected number of real zeros for random linear combinations of
orthogonal polynomials. It is well known that Kac polynomials, spanned by
monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$
expected real zeros in terms of the degree $n$. If the basis is given by the
orthonormal polynomials associated with a compactly supported Borel measure on
the real line, or associated with a Freud weight defined on the whole real
line, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real
zeros. We prove that the same asymptotic relation holds for all random
orthogonal polynomials on the real line associated with a large class of
weights, and give local results on the expected number of real zeros. We also
show that the counting measures of properly scaled zeros of these random
polynomials converge weakly to either the Ullman distribution or the arcsine
distribution.
