Given a joint probability density function of N real random variables, fxjgNj D1, obtained from the eigenvector–eigenvalue decomposition of N N random matrices, one constructs a random variable, the linear statistics, defined by the sum of smooth functions evaluated at the eigenvalues or singular values of the random matrix, namely,PNj D1 F.xj/. For the joint PDFs obtained from the Gaussian and Laguerre ensembles, we compute, in this paper, the moment-generating function Eˇ.exp. Pj F.xj///, where Eˇ denotes expectation value over the orthogonal (ˇ D 1) and symplectic (ˇ D 4) ensembles, in the form one plus a Schwartz function, none vanishing over R for the Gaussian ensembles and RC for the Laguerre ensembles. These are ultimately expressed in the form of the determinants of identity plus a scalar operator, from which we obtained the large N asymptotic of the linear statistics from suitably scaled F. /. Copyright © 2016 John Wiley & Sons, Ltd. Keywords: random matrices; linear spectra