  In this paper, we study the large $N$ behavior of the moment-generating
function (MGF) of the linear statistics of $N\times N$ Hermitian matrices in
the Gaussian unitary, symplectic, orthogonal ensembles (GUE, GSE, GOE) and
Laguerre unitary, symplectic, orthogonal ensembles (LUE, LSE, LOE) at the edge
of the spectrum. From the finite $N$ Fredholm determinant expression of the MGF
of the linear statistics, we find the large $N$ asymptotics of the MGF
associated with the Airy kernel in these Gaussian and Laguerre ensembles. Then
we obtain the mean and variance of the suitably scaled linear statistics. We
show that there is an equivalence between the large $N$ behavior of the MGF of
the scaled linear statistics in Gaussian and Laguerre ensembles, which leads to
the statistical equivalence between the mean and variance of suitably scaled
linear statistics in Gaussian and Laguerre ensembles. In the end, we use the
Coulomb fluid method to obtain the mean and variance of another type of linear
statistics in GUE, which reproduces the result of Basor and Widom.
