Suppose you have some random vector $\displaystyle \bold{X} = (X_1, \dots, X_n) $ with a density function $\displaystyle f(x_1, \dots, x_n) $. Suppose you want to compute the expected value of some function of the random vector (e.g. $\displaystyle E[g(\bold{X})] $). We know that

$\displaystyle E[g(\bold{X})] = \int \int \cdots \int g(x_1, \dots, x_n) f(x_1, \dots, x_n) dx_1 \cdots dx_n $

What is a good way of developing approximation methods for computing $\displaystyle E[g(\bold{X})] $? If $\displaystyle g(\bold{X}) = \bold{X} $ then we are just computing the expected value of $\displaystyle \bold{X} $. So then we can reduce this question (in this case) to what are some good ways of approximating the expected value of a random vector?