Given a joint probability density function (jpdf) of N real random variables, {xj} N j=1, 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, PN j=1 F(xj ). Linear statistics is an ubiquitous statistical characteristic in random matrix theory. For the jpdfs obtained from the Gaussian and Laguerre ensembles, we compute, in this thesis the moment-generating function Eβ(exp(−λ P j F(xj ))), where Eβ denotes expectation value over the Orthogonal (β = 1), Unitary (β = 2) and Symplectic (β = 4) ensembles, in the form of one plus a Schwartz function, non vanishing over R for the Gaussian ensembles and over R + 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(·). For Hermitian random matrix ensembles, we show that if F(·) is a polynomial of degree K, then the variance of tr F(M) or PN j=1 F(xj ), is of the form, PK n=1 n(dn) 2 , and dn is related to the expansion coefficients cn of the polynomial F(x) = PK n=0 cnPbn(x), where Pbn(x) are polynomials of degree n, orthogonal with respect to the weights √ 1 (b−x)(x−a) , p (b − x)(x − a), √ (b−x)(x−a) x , (0 < a < x < b), √ (b−x)(x−a) x(1−x) , (0 < a < x < b < 1), respectively. ii Decl