  We discuss quantum algorithms that calculate numerical integrals and
descriptive statistics of stochastic processes. With either of two distinct
approaches, one obtains an exponential speed increase in comparison to the
fastest known classical deterministic algorithms and a quadratic speed increase
in comparison to classical Monte Carlo (probabilistic) methods. We derive a
simpler and slightly faster version of Grover's mean algorithm, demonstrate how
to apply quantum counting to the problem, develop some variations of these
algorithms, and show how both (apparently quite different) approaches can be
understood from the same unified framework. Finally, we discuss how the
exponential speed increase appears to (but does not) violate results obtained
via the method of polynomials, from which it is known that a bounded-error
quantum algorithm for computing a total function can be only polynomially more
efficient than the fastest deterministic classical algorithm.
