常用离散型随机变量的高阶原点矩

对离散型随机变量的高阶矩进行了研究,给出了几类离散型随机变量的高阶原点矩的统一递推公式,得到了离散型随机变量的高阶原点矩的形式特征.     

用伯努利试验研究离散型随机变量的分布

将常见的离散型随机变量的分布用一个与伯努利试验有关的题目加以整合,揭示了怎样由伯努利试验构造离散型随机变量的分布．并举出了基于伯努利试验的离散型随机变量的分布在解决相关问题中的例子．实例表明用伯努利试验研究离散型随机变量的分布及相关问题．会起到事半功倍的效果．     

离散型随机变量密度函数的定义与探讨

使用“δ函数”定义离散型随机变量的密度函数，寻求离散型随机变量与连续型随机变量的统一处理方法．基于离散型随机变量密度函数的定义．其一维随机变量函数的密度函数以及多维随机变量的边缘密度等，均可直接利用连续型随机变量的相关结论．     

图解常用离散型随机变量

在概率论的学习中,一个重要章节就是常用的离散型随机变量的学习.离散型随机变量包括伯努利分布,二项分布,泊松分布,几何分布,超几何分布和负二项分布等等.在本文中,首先借助时间流的图形表达,从伯努利试验次数和成功次数角度区分其中的一些常用变量;其次通过一个流程图的方式梳理这些常用的离散型随机变量的定义.本文的目的在于,基于常规的离散型随机变量的分布律等介绍之余,首次尝试从不同的比较汇总角度,借助图表方法对常用的离散型随机变量进行梳理和总结,起到区分变量的差异,加强对常用离散型随机变量概念的理解.     

一个离散型随机变量数学期望的几种简捷求法

直接利用期望定义来求离散型随机变量的数学期望,有时计算比较困难.利用条件数学期望、随机变量的和式分解、对称性,分别给出了一个离散型随机变量数学期望的几种求法.     

求离散型随机变量的均值与方差的四种方法

<正>离散型随机变量的均值与方差是高考的热点.均值或数学期望，反映了离散型随机变量取值的平均水平，方差或标准差反映了随机变量取值偏离于均值的平均程度，均值与方差是随机变量的两个重要的数字特征.求离散型随机变量的均值与方差有定义分析法、性质求解法、图象转化法、特殊分布法四种方法.     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本文研究了离散型随机变量次序统计量的分布矩阵的对称性 ,获得了二个定理 .定理 1　服从等概率二点分布或等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵 .定理 2　取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性 .     

离散型随机变量数学期望的几种巧妙算法

利用定义求解离散型随机变量的数学期望有时显得非常复杂,本文给出了三种巧妙计算离散型随机变量数学期望的方法:对称性法、随机变量分解法、公式演变法.计算过程非常简洁,达到了简化计算的目的.     

关于离散型随机变量次序统计量分布矩阵对称性猜想的探索

本研究了离散型随机变量次序统计量的分布矩阵的对称性,获得了二个定理。定理1服从等概率二点分布等概率三点分布的离散型随机变量的次序统计量的分布矩阵是对称矩阵。定理2取值有限且等概率的离散型随机变量的次序统计量的分布矩阵具有中心对称性。     

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.     

离散指数分布序列的强偏差与强逼近

本文引入离散指数分布概念,建立了关于离散型指数分布序列的强偏差定理和强大数定律.同时,得到离散指数分布序列对连续指数分布序列的强逼近.     

Cycle-transitive comparison of independent random variables

The discrete dice model, previously introduced by the present authors, essentially amounts to the pairwise comparison of a collection of independent discrete random variables that are uniformly distributed on finite integer multisets. This pairwise comparison results in a probabilistic relation that exhibits a particular type of transitivity, called dice-transitivity. In this paper, the discrete dice model is generalized with the purpose of pairwisely comparing independent discrete or continuous random variables with arbitrary probability distributions. It is shown that the probabilistic relation generated by a collection of arbitrary independent random variables is still dice-transitive. Interestingly, this probabilistic relation can be seen as a graded alternative to the concept of stochastic dominance. Furthermore, when the marginal distributions of the random variables belong to the same parametric family of distributions, the probabilistic relation exhibits interesting types of isostochastic transitivity, such as multiplicative transitivity. Finally, the probabilistic relation generated by a collection of independent normal random variables is proven to be moderately stochastic transitive.     

Truncated random variables with application to random surfaces

There has been considerable interest in obtaining discrete results for random surfaces. Standard results have been published in journals of physics or engineering which have emphasised the applications. This paper gives a detailed background of the mathematical methods needed so that the central connection, namely truncated random variables, between these standard results can be understood. Distributions of discrete peak measures are obtained from the distributions of discrete profile measures of a random Gaussian surface by applying results for the distributions of truncated random variables. This enable the moments to be obtained from known results for the truncated distributions.     

有关数学期望的优化问题

讨论随机变量函数的数学期望的求解及其应用问题.针对连续型和离散型随机变量,实例说明有关数学期望的一类随机优化模型的决策变量的求解问题,指出在离散情形下相关问题的传统解法容易漏解,并给予修正.     

一类非离散非连续的随机变量的概率分布

本文定义了一类非离散非连续的随机变量的概率分布,使其概率分布与离散型和连续型随机变量的概率分布表示保持一致,并举例求出考研中涉及过的该类型的随机变量的概率分布.     

Properties of graphical regression models for multidimensional categorical data

We propose a two-component graphical chain model, the discrete regression distribution, where a set of discrete random variables is modeled as a response to a set of categorical and continuous covariates. The proposed model is useful for modeling a set of discrete variables measured at multiple sites along with a set of continuous and/or discrete covariates. The proposed model allows for joint examination of the dependence structure of the discrete response and observed covariates and also accommodates site-to-site variability. We develop the graphical model properties and theoretical justifications of this model. Our model has several advantages over the traditional logistic normal model used to analyze similar compositional data, including site-specific random effect terms and the incorporation of discrete and continuous covariates.     

既非离散又非连续的随机变量

讨论既非离散又非连续的随机变量,总结这类随机变量分布函数的特点,由此可进行这类随机变量类型的判定.通过举例给出既非离散又非连续型随机变量数学期望的求法.     

On the transitivity of the comonotonic and countermonotonic comparison of random variables

A recently proposed method for the pairwise comparison of arbitrary independent random variables results in a probabilistic relation. When restricted to discrete random variables uniformly distributed on finite multisets of numbers, this probabilistic relation expresses the winning probabilities between pairs of hypothetical dice that carry these numbers and exhibits a particular type of transitivity called dice-transitivity. In case these multisets have equal cardinality, two alternative methods for statistically comparing the ordered lists of the numbers on the faces of the dice have been studied recently: the comonotonic method based upon the comparison of the numbers of the same rank when the lists are in increasing order, and the countermonotonic method, also based upon the comparison of only numbers of the same rank but with the lists in opposite order. In terms of the discrete random variables associated to these lists, these methods each turn out to be related to a particular copula that joins the marginal cumulative distribution functions into a bivariate cumulative distribution function. The transitivity of the generated probabilistic relation has been completely characterized. In this paper, the list comparison methods are generalized for the purpose of comparing arbitrary random variables. The transitivity properties derived in the case of discrete uniform random variables are shown to be generic. Additionally, it is shown that for a collection of normal random variables, both comparison methods lead to a probabilistic relation that is at least moderately stochastic transitive.     

A Characterization of Joint Distribution of Two-Valued Random Variables and Its Applications

We obtain an explicit representation for joint distribution of two-valued random variables with given marginals and for a copula corresponding to such random variables. The results are applied to prove a characterization of r-independent two-valued random variables in terms of their mixed first moments. The characterization is used to obtain an exact estimate for the number of almost independent random variables that can be defined on a discrete probability space and necessary conditions for a sequence of r-independent random variables to be stationary.