# statistics: expected value for the eigenvalue of the random orthogonal matrix

I am interested in finding the expected value of the product of two eigenvalues ​​with respect to the GOE set.

So I have a chance $$n times n$$ Matrix with all the elements diagonally. $$mathcal {N} (0,1)$$ and diagonally $$mathcal {N} (0.2)$$ all independent is well known that it is possible to evaluate the explicit p.d.f. For the eigenvalues ​​of this type of matrix:
$$f ( lambda) d lambda = n! Delta ( lambda) e ^ {- frac 1 2 sum lambda_j ^ 2}$$

But with this formulation now I can not calculate, for example, $$mathbb {E}[lambda_j]$$ or $$mathbb {E}[lambda_jlambda_i]$$, Is there any technique to evaluate these integrals? Is it possible to use the asymptotic density function (that is, the law of the semicircle) to calculate them or at least have an estimate with a remainder in terms of the size of the matrix?