Lower bounds for sampling algorithms for estimating the average

We show lower bounds on the number of sample points and on the number of coin tosses used by general sampling algorithms for estimating the average value of functions over a large domain. The bounds depend on the desired precision and on the error probability of the estimate. Our lower bounds match upper bounds established by known algorithms, up to a multiplicative constant. Furthermore, we give a non-constructive proof of existence of an algorithm that improves the known upper bounds by a constant factor.