条件数学期望与随机变量独立性的一个充要条件

随机试验的独立性、随机事件的独立性、随机变量的独立性均是概率统计中的重要概念,不少学者都在这些方面有所讨论.本文作者就二维离散形随机向量(ξ,η)中两个分量ξ与η的相互独立性展开讨论.先是证明了三个引理,其中引理1在一般概率论教科书中均有介绍,但为使读者方便,作者也作了证明.利用三个引理,作者找到随机变量独立性的一个充要条件.     

连续随机变量的随机独立性与回归独立性

回归独立性是指给定随机变量 X时 ,随机变量 Y的条件期望 E( Y|X)不依赖于 X.前人讨论了离散型随机变量回归独立性与随机独立性的关系 ,得到了二者等价的充分必要条件 .对连续型随机变量的情形加以讨论 ,获到了二者等价的几个充分必要条件 ,并说明在统计分析中的应用 .     

二元Freund型指数分布的独立性和不相关性

本文讨论二元Freund型指数分布的独立性和不相关性,分别获得了两个服从二元Freund型指数分布随机变量相互独立及不相关的充分必要条件;并得到了其相关系数的精确表达式,证明了在对称情形其相关系数落入负三分之一与一之间,文末证明了X,Y之间的渐近独立性.     

西摩松线的一个性质

大家知道 :三角形外接圆上任一点在三边所在直线上的射影共线 ,这条直线称做该点对于三角形的西摩松线 (Simson) .本文将给出关于三角形西摩松线的一个新性质 .定理　三角形的三个外角平分线与其外接圆交点的西摩松线共点 .已知　如图 1,在△ ABC(AB≥ AC)中 ,X、Y、Z分别是△ ABC三个外角∠ DAB、∠ ABE、∠ BCF的平分线 AX、BY、CZ与△ ABC外接圆的交点 ,且点 Xi、Yi、Zi(i =1,2 ,3)分别是点 X、Y、Z在直线 AB、BC、CA上的射影 .求证　直线 X1 X2 X3 、Y1 Y2 Y3 、Z1 Z2 Z3 三线共点 .先给出一个引理 :引理 [1 ]　…     

三角形旁切圆切点的一个性质

定理　设△ABC的旁切圆⊙Ia、⊙Ib、⊙Ic 分别切BC、CA、AB于点X、Y、Z .过YZ和BC的中点X1和D作一直线X1D ,及类似的直线Y1E和Z1F(如图 1) .则X1D、Y1E、Z1F三线共点且该点恰为△DEF的内心 .先给出下面的引理 .引理 1[1] 　分别过三角形三边中点的三条周界平分线交于一点 ,这一点称为第二等周中心 (证明略 ) .图 1　　　　　　图 2引理 2　若四边形的一组对边相等 ,则相等的这一组对边交角的平分线必平行于另一组对边中点的连线 .证明　如图 2 ,设四边形ABCD中 ,AD=BC ,E、F分别为AB、CD的中点 ,AD、BC的延长线交于点…     

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本文研究了独立随机变量之和的绝对矩的几个性质, 其中包括$\ep|X+Y|-\ep|X-Y|$的表达式, 这里$X$和$Y$是相互独立的随机变量.     

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.     

由域Z3上随机变量序列构造的剩余类环Z9上随机变量序列的统计性质分析

本文利用域Z3上的两条随机变量序列通过映射得到剩余类环Z9上的一条随机变量序列,证明了当Z3上的两条随机变量序列具有良好的统计特性(均匀性、独立性、严平稳性、马氏性)时,Z9上的随机变量序列也具有相应的良好统计特性.     

二元Arnold-Strauss型指数分布的条件指数性及渐近独立性

本文讨论二元Arnold-Strauss型指数分布的条件指数性及渐近独立性,证明了给定X关于Y的条件密度和给定Y关于X的条件密度都是指数分布密度,求出了用于预报的条件概率;并证明了X,Y之间的渐近独立性.另外,讨论了它的识别性,若已知可识最小值的分布密度时,所有参数皆可识别.     

Z=X+Y概率密度的拉氏变换解法

本文根据拉氏变换的卷积性质 ,给出了某些情形下 ,随机变量 Z=X+Y概率密度的一种简易的拉氏解法     

离散型随机变量的几种独立性及相互关系

随机变量的独立性在概率论中有着十分重要的意义.本文给出了离散型随机变量与离散型随机向量相互独立的概念,条件独立的概念,以及几种独立性的相互关系.     

三变量独立性的两个结论

本文给出了三随机变量相互独立与条件独立的两个结论．其中结论２表明，三变量如果两两条件独立，则三变量一定相互独立．     

Logical and algorithmic properties of stable conditional independence

The logical and algorithmic properties of stable conditional independence (CI) as an alternative structural representation of conditional independence information are investigated. We utilize recent results concerning a complete axiomatization of stable conditional independence relative to discrete probability measures to derive perfect model properties of stable conditional independence structures. We show that stable CI can be interpreted as a generalization of Markov networks and establish a connection between sets of stable CI statements and propositional formulas in conjunctive normal form. Consequently, we derive that the implication problem for stable CI is coNP-complete. Finally, we show that Boolean satisfiability (SAT) solvers can be employed to efficiently decide the implication problem and to compute concise, non-redundant representations of stable CI, even for instances involving hundreds of random variables.     

The essential equivalence of pairwise and mutual conditional independence

For a large collection of random variables, pairwise conditional independence and mutual conditional independence are shown to be essentially equivalent — i.e., equivalent to up to null sets. Unlike in the finite setting, a large collection of random variables remains essentially conditionally independent under further conditioning. The essential equivalence of pairwise and multiple versions of exchangeability also follows as a corollary. Our result relies on an iterated extension of Bledsoe and Morse's completion of the product of two measure spaces. Part of this work was done while Peter Hammond was visiting the National University of Singapore in March–April 2004. The final version was completed while Yeneng Sun was on sabbatical leave at the University of Illinois at Urbana–Champaign and Stanford University in October 2004 – May 2005.     

Conditional versions of limit theorems for conditionally associated random variables

Relation between association and conditional association is answered, several examples show that the association of random variables does not imply the conditional association, and vice versa. Several fundamental properties of conditional associated random variables are developed, which extend the corresponding ones under the non-conditioning setup. By means of these properties, some conditional Hájek-Rényi type inequalities, a conditional strong law of large numbers and a conditional central limit theorem stated in terms of conditional characteristic functions are established, which are conditional versions of the earlier results for associated random variables, respectively. In addition, some lemmas in the context are of independent interest.     

Some properties of record values coming from the geometric distribution

The independence between spacings of record values and of individual record values is well known when the records come from a geometric distribution. Here we examine the form a function of two record values must have if we require independence from a lower record value. Also similar questions are examined in relation to conditional expectation and conditional variance.     

Independence for full conditional probabilities: Structure,factorization, non-uniqueness,and Bayesian networks

This paper examines concepts of independence for full conditional probabilities; that is, for set-functions that encode conditional probabilities as primary objects, and that allow conditioning on events of probability zero. Full conditional probabilities have been used in economics, in philosophy, in statistics, in artificial intelligence. This paper characterizes the structure of full conditional probabilities under various concepts of independence; limitations of existing concepts are examined with respect to the theory of Bayesian networks. The concept of layer independence (factorization across layers) is introduced; this seems to be the first concept of independence for full conditional probabilities that satisfies the graphoid properties of Symmetry, Redundancy, Decomposition, Weak Union, and Contraction. A theory of Bayesian networks is proposed where full conditional probabilities are encoded using infinitesimals, with a brief discussion of hyperreal full conditional probabilities.     

集值条件期望的收敛定理

本文给出了集合序列弱极限的表示定理，得到了随机集列关于σ－域流的条件期望序列在弱收敛意义的Fatou型引理和控制收敛定理。