Covariance of the Number of Real Zeros of a Random Algebraic Polynomial with Binomial Elements

Abstract

This paper provides a formula to be used for obtaining the variance of the number of real zeros of random algebraic polynomial . The expected number of real zeros of this type of polynomial is known. An easy modification of this formula leads to a formula for the covariance for the number of real zeros in any two disjoint intervals. Using the latter, we show the covariance of the number of real zeros, in any two disjoint interval that can be obtained. To this end, we assume a normal standard distribution for the coefficients a j 's, j = 0, 1, 2,…, n. Although we give a formula for the variance, the evaluation of the asymptotic value for the variance remains our main task for future work.