AI 中文总结
本文研究Poisson多面体及Crofton多面体体积、表面积和面积的分布，提出混合分布模型并引入广义伽马分布，通过模拟估计参数，首次精确给出面数概率。
AI 中文摘要
本文介绍了关于Poisson多面体和Crofton多面体的体积与表面积分布的新发现。同时，本文还考虑了平面上Poisson多边形和Crofton多边形面积的分布。本文提出了将这些凸体泛函的分布建模为混合分布，并强调广义伽马分布作为混合成分的必要性。最后，基于这些随机多面体的大量随机模拟，对模型参数进行了估计并选择了模型。这包括首次给出了Poisson多面体和Crofton多面体的面数概率的非常精确的估计，因为这些概率在分布模型中用作混合权重。
英文摘要
This note presents new findings on the distribution of the volume and surface area of both the Poisson polyhedron and the Crofton polyhedron. It also considers the distribution of the area of the Poisson polygon and the Crofton polygon in the plane. This note motivates the modeling of the distributions of these convex body functionals as mixture distributions and the essential inclusion of the generalized gamma distribution therein as mixture components. Finally, model parameters are estimated and models are selected based on extensive stochastic simulations of these random polytopes. This includes that for the first time, this note gives very accurate estimates of the side number probabilities for both the Poisson polyhedron and the Crofton polyhedron, as these are used as mixing weights in the distribution models.
Comments14 pages
