Journal of Applied Mathematics and Stochastic Analysis
Volume 10 (1997), Issue 3, Pages 257-264
Real zeros of a random polynomial with Legendre elements
University of Ulster, Department of Mathematics, Co. Antrim, Jordanstown BT37 0QB, United Kingdom
Received 1 May 1996; Revised 1 December 1996
Copyright © 1997 K. Farahmand. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Let be a sequence of normalized Legendre polynomials orthogonal with respect to the interval . The asymptotic
estimate of the expected number of real zeros of the random polynomial
where , 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.