Past exam of the chemistry course of the year 4 of the natural sciences course of the University of Cambridge / 2022 / Paper 1 / 7 / c / Solution

The constant density is reasonable when the integrand has comparable magnitude throughout the interval. It is inefficient for a narrow peak because most samples contribute little while rare samples carry large weights.

For independent importance sampling,A good density has support wherever $f\ne 0$, is easy to sample and evaluate, and resembles $|f|$. Subject to normalization, $q\propto |f|$ minimizes this variance. For nonnegative $f$, the ideal normalized choice makes every sample contribution equal, although its normalization is generally the unknown integral itself. Uniform weighting is therefore acceptable for a flat integrand, not universally suitable.