I have to approximate an integral with respect to three random variables, two of which are bivariate normally distributed with a nonzero correlation coefficient, the third is simply independently normally distributed. A more formal problem formulation would be: Let eta, u and epsilon be three normally distributed random variables with probability density function p(eta, u, epsilon). The correlation coefficient between eta and u is positive ! The distribution of epsilon is independent of the other distribution. Let f(eta, u, epsilon) be some analytically known function. Finally let
U(f(eta, u, epsilon)) be be a numerically tabulated (i.e. I’ve got a grid of U-values for specific f-values), but analytically unknown function. Now I need to approximate E[U(f(eta, u, epsilon)].

I don’t really know how to handle the problem, i.e. how to approximate the integral.
I thought maybe I could get some discrete values for the three random variables, use them to calculate f(.), use f(.) to look up the corresponding U(.) , multiply the U(.) with the probability for epsilon and with the probability for (u, eta) and then sum it all up.

Is there a more speedy / accurate way for this integration? I don’t know if it makes sense to use equally spaced discrete values (picked arbitrarily) for this kind of problem???

I would greatly appreciate any suggestions!!