  We study $L_q$-approximation and integration for functions from the Sobolev
space $W^s_p(\Omega)$ and compare optimal randomized (Monte Carlo) algorithms
with algorithms that can only use iid sample points, uniformly distributed on
the domain. The main result is that we obtain the same optimal rate of
convergence if we restrict to iid sampling, a common assumption in learning and
uncertainty quantification. The only exception is when $p=q=\infty$, where a
logarithmic loss cannot be avoided.

Numerical Analysis;Numerical Analysis