  Given a joint probability density function of $N$ real random variables,
$\{x_j\}_{j=1}^{N},$ obtained from the eigenvector-eigenvalue decomposition of
$N\times N$ 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, $\sum_{j=1}^{N}F(x_j).$ For
the jpdfs obtained from the Gaussian and Laguerre ensembles, we compute, in
this paper the moment generating function $\mathbb{E}_{\beta}({\rm
exp}(-\lambda\sum_{j}F(x_j))),$ where $\mathbb{E}_{\beta}$ denotes expectation
value over the Orthogonal ($\beta=1$) and Symplectic ($\beta=4)$ ensembles, in
the form one plus a Schwartz function, none vanishing over $\mathbb{R}$ for the
Gaussian ensembles and $\mathbb{R}^+$ 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 $F(\cdot).$
