Weighted Monte Carlo calculations requiring a uniform sampling of the problem-space can suffer from diminished statistical significance because many, if not most, of the randomly-chosen sampling points contribute only slightly to the desired result. Their contribution is reduced in size due the variable-size of the weighting terms. In contrast, none of the randomly-chosen points which are favored by variable size weighting terms will have their statistical significance enhanced beyond that of just one random point in the Monte Carlo sampling. A Monte Carlo analysis was used in earlier work to verify both Gauss' Law and Newton's Shell Theorem. Both examples suffered from statistical difficulties since each Monte Carlo sampling point has a weight inversely proportional to the square of the distance between source and field points. The present work analyzes the diminished significance in weighted Monte Carlo for the specific example of Newton's Shell Theorem, describing the geometry in terms of closest approach distance of the spherical mass shell to the field point. Binomial Statistics is used to remedy this diminished statistical significance by providing a prescription for increasing the value of the Monte Carlo sample size needed to assure that the chosen precision remains invariant as the mass-shell geometry is changed.
McCloskey, Sue Ellen; Hall, William C.; and Braithwaite, Wilfred J.
"Applying Binomial Statistics to Weighted Monte Carlo,"
Journal of the Arkansas Academy of Science: Vol. 52
, Article 12.
Available at: https://scholarworks.uark.edu/jaas/vol52/iss1/12