Let T0∗(x),T1∗(x),…,Tn∗(x) be a sequence of normalized Legendre polynomials orthogonal with respect to the interval (−1,1). The asymptotic
estimate of the expected number of real zeros of the random polynomial
g0T0∗(x)+g1T1∗(x)+…+gnTn∗(x) where gj, j=1,2,…,n are independent
identically and normally distributed random variables with mean zero and
variance one is known. The present paper considers the case when the
means and variances of the coefficients are not all necessarily equal. It is
shown that in general this expected number of real zeros is only dependent
on variances and is independent of the means.