统计模拟在几何概率问题中应用的注解

A note on statistics simulation for geometric probability problems

设D为Rs上的一个紧集,X为D上的一个随机覆盖过程的统计量.由于问题复杂,X的均值、方差、分布函数均没有解析表达式.统计模拟可以帮助我们找到它们的近似解.为了在D上做统计模拟,需要D的代表点.产生代表点的不同方法,会影响统计模拟的结果.若D不是一个矩形,如何选择合适的代表点至关重要.文献中研究了一个在单位圆上的随机覆盖问题,提出在单位圆上产生代表点的四种方法,并对这四种方法给予评估.本文考虑两个随机圆的随机覆盖问题,给出覆盖面积的理论公式,使比较四种产生代表点的方法有一个基准.我们的研究结果和文献中的结论一致,并发现其中两种方法使覆盖面积均值的估计有偏,且有较大的方差,这是一个新的结果.本文进一步指出覆盖面积的分布可由β分布来拟合.

Let D be a geometric compact domain in R2 and X be a statistic based on a random coverage process on D. Due to the problem complexity there are no analytic formulas for the mean, variance and distribution function of X. A statistical simulation can help us find approximation solutions to them. To use a statistical simulation the result is effected by a set of representative points for D. If D is not a rectangle, how to choose a suitable set of representative points is a very important issue. In this paper we discuss a real-life case study that is a geometric coverage problem on the unit cycle covered by m random cycles, which had been discussed in the literature. They proposed four methods of generating a set of representative points of D and gave some evaluation on these four methods. In this paper we consider the case of two random cycles and derive the analytic formulas for the coverage area so that we have a benchmark for comparing the four methods. Our conclusions are consistent with that given in the literature. Furthermore, we find that two methods imply biased estimator of the coverage area with a larger variance. The paper also shows that the distribution of the coverage area can be approximated by a β distribution.