if we are given a finite number N of points drawn from a probability distribution, expectation can be approximated as a finite sum over these points: E[f]=(1/N)(summation of f(x) over these N points).

comparing this to the actual calculation of E[f]=summation of p(x)f(x), won't the difference between the actual value and approximate value be a lot in cases where p(x) varies a lot?