圆内两随机点距离的概率分布

研究了圆内两个均匀分布随机点之间距离的概率分布.利用几何概率推导出了概率分布函数、概率密度函数和数学期望的解析表达式.基于R语言,对单位圆内随机点之间距离进行了随机模拟,模拟结果与理论分析结果互为印证.     

随机针偶与凸体K相交的几何概率问题

本文研究了随机针偶与凸体K相交的几何概率,利用有向直线偶的运动不变密度公式,获得了针偶的运动不变密度公式,从而进一步得到随机针偶与凸体K相交且针偶的交点属于K的几何概率.     

几何分布中两类问题的说明

在贝努里(Bemoulli)试验中，事件A发生的概率为p，若以ξ记A首次出现时所需的试验次数，则ξ是随机变量，它的所有可能取值为1，2，3，…，n，…，且概率函数为     

随机环境中分枝过程灭绝概率的性质

本文给出了随机环境中分枝过程概率母函数以及灭绝概率的性质.     

一类几何概率问题的推广

文从著名的蒲丰投针问题出发,得到凸ｎ边形与一组等距离的平行线相交的概率公式．     

赌博破产概率及其随机模拟试验

讨论了赌博问题中的最终破产概率,并给出了破产概率的随机模拟计算流程和一个具体例子的数值模拟结果.计算结果表明,由此方法得到破产概率的估计值与理论值的误差很小.最后,通过随机模拟给出游戏结束的平均次数.     

超几何分布的数字特征和概率计算的探讨

用定义法、性质法、概率母函数法三种方法探索了超几何分布的数学期望和方差的求法,同时又给出了超几何分布、二项分布、泊松分布和正态分布之间的近似关系,从而解决了超几何分布的概率计算问题.     

搜索型随机格斗命中次数的分布与毁伤概率

本文研究了搜索型随机格斗模型，采用在时间[0，t]内射击命中目标次数的概率分布与毁伤概率指标来评定射击效果。文章对射击时间间隔随机变量服从的分布采用两种假设t指数分布与一般分布．对指数分布的假设，文章导出的结果比较完善；对一般分布的假设，文章导出了计算射击命中次数概率分布的计算公式。     

大数据条件下概率统计教学的两点思考

本文借助"概率元"工具讨论了高维、分量类型不一致的随机变量的分布描述方法,借助"随机模拟"技术和贝叶斯方法讨论了过程数据的处理方法,希望借此给当下统计教学一些补充,突出统计学在大数据时代的地位.     

The Distribution of the Rational Points on the Unit Circle

In the paper, the explicit form of distribution function for the lengths of arcs connecting neighbouring rational points on the unit circle whose denominators do not exceed given value, is given.     

一类两个随机变量函数的分布

给出了随机变量X,y分别是离散、连续不同类型时,其函数Z=g（X,y）分布的一般求解方法和应用举例．     

Expected Distances between Two Uniformly Distributed Random Points in Rectangles and Rectangular Parallelpipeds

In this paper, the expected distance between two uniformly distributed random points in a rectangle or in a rectangular parallelepiped is computed under three different metrics: the Manhattan metric, the Euclidean metric, and the Chebychev metric.     

Calculation of the Distance betweeen Two Random Sets

Two independent random sets are formed by finite systems of independent random variables uniformly distributed on the unit circle. A formula for the distribution of the distance between the sets is derived.     

Spherical Distribution of 5 Points with Maximal Distance Sum

In this paper, we consider the problem of spherical distribution of 5 points, that is, how to configure 5 points on the unit sphere such that the mutual distance sum is maximal. It is conjectured that the sum of distances is maximal if the 5 points form a bipyramid distribution with two points positioned at opposite poles of the sphere and the other three positioned uniformly on the equator. We study this problem using interval methods and related techniques, and give a computer-assisted proof.     

Fitting a Set of Points by a Circle

Given a set of points S={p ₁ ,. . ., p _n } in Euclidean d -dimensional space, we address the problem of computing the d -dimensional annulus of smallest width containing the set. We give a complete characterization of the centers of annuli which are locally minimal in arbitrary dimension and we show that, for d=2 , a locally minimal annulus has two points on the inner circle and two points on the outer circle that interlace anglewise as seen from the center of the annulus. Using this characterization, we show that, given a circular order of the points, there is at most one locally minimal annulus consistent with that order and it can be computed in time O(n log n) using a simple algorithm. Furthermore, when points are in convex position, the problem can be solved in optimal Θ(n) time. Received June 25, 1997, and in revised form March 5, 1998.     

对两点关于圆周对称定义的讨论

本文给出了两点关于圆周对称的一种新定义，克服了［１］，［２］所给定义的不严密性。     

例说独立性与不相关性的关系——兼说高等数学与概率统计的关系

本文讨论独立性与不相关性的关系,除了随机变量之间的独立性必定导致二个随机变量之间的不相关性外,以几个服从二元二点分布、二元均匀分布、二元指数分布、二元正态分布的二个随机变量之间的独立性与不相关性作为例子:说明独立性与不相关性之间的其他三种情形;不难发现,刻划独立性的是函数关系,刻划不相关性的是数字关系;由此,同时讨论高等数学与概率统计的关系,也列出了高等数学与概率统计之间的四种情形.     

Self-Similarity of Some Sequences of Points on a Circle

The self-similarity and periodicity properties are proved for the derivatives d^mO₀ of the sequences O₀, which are obtained by shifting the unit circle by the arc . Bibliogrhaphy: 5 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 302, 2003, pp. 81–95.