概率分布的实函数实现问题

基于一维随机变量,通过阐释概率分布实函数实现的内在本质,给出了概率分布实函数实现的一个充分必要条件,得到分布函数族及其连续性特征,揭示出概率论中分布函数定义所蕴含的合理性和深刻性.     

关于大偏差概率的一个界

研究得到了关于随机和S(t)=∑N(t)i=1Xi,t≥0大偏差的幂的一个界,其中(N(t))t≥0是一族非负整值随机变量,(Xn)n∈N是独立同分布的随机变量,其共同的分布函数是F与(N(t))t≥0独立.本结论是在假设分布函数F的右尾属于ERV族的情况下得到的.     

在独立而不同分布的随机截断下Kaplan-Meier估计在半直线上的弱收敛

假定t_1,t_2,…t_n是一族独立同分布的随机变量,具有分布函数G,并且被另一族随机变量y_1,y_2,…,y_n所截断。我们只能观察到Z_t=min(t_i,y_i)一种估计G的办法是用Kaplan-Meier估计(?)_n,本文讨论了{y_i}独立而不同分布的情况,证明了√(?)((?)_n-G)在半直线上的弱收敛。     

混合型随机变量数字特征的计算

利用分布函数的广义反函数,将随机变量表示为均匀分布随机变量的函数,然后用随机变量函数的数学期望公式可计算原随机变量的数字特征,该方法适用于所有类型的随机变量.最后,利用四种不同方法计算了一个混合型随机变量的数字特征,说明了这些方法的异同.     

重尾随机变量加权和一致尾等价关系成立的条件

已有文献指出在次指数族重尾损失的情形下,独立同分布随机变量加权和的一致尾等价关系成立.文中主要在随机变量独立但不同分布的情况下,得到了上述尾等价关系一致成立时分布函数所应满足的一些条件.     

二维连续型随机变量函数的分布密度的计算

二维连续型随机变量函数的密度函数的计算既是概率论教学中的一个重点,又是一个难点.本文介绍了一般二维连续型随机变量函数的分布密度的计算方法,并给出了一个新的方法——密度函数转化法.     

L族END的多元部分和的精确大偏差下界

从保险的实际出发,研究服从长尾分布族(L族)上的多元风险模型中随机变量序列的部分和的精确大偏差,其中假设随机变量序列是一列延拓负相依(END)的、同分布的随机变量序列,利用基于求L族的精确大偏差的方法得到了随机变量部分和的渐近下界.     

数据缺失机制不明确时的分布函数估计

研究了在数据缺失机制不明确时如何估计随机变量Y的分布函数FY(y),该问题不同于可以用参数模型刻画数据缺失机制时的情形,考虑到此时可能出现不可识别现象,获取一些辅助信息是必要的.借助一个可以完全观察到的随机变量X提供必须的辅助信息,构造了随机变量Y的分布函数Fy(y)的估计量,并研究了它的大样本性质.     

次指数随机变量最大和的大偏差概率及其在有限时间破产概率中的应用

对于由独立同分布的标准均匀分布随机变量中心化的次指数随机变量序列,对于其部分和的最大值 建立了一个大偏差概率的渐近关系.该结果扩展了Korshunov相应的结论. 作为应用, 将Tang的结果,即关于有限时间破产概率的一致渐近估计,由一致变化分布族推广到了整个强次指数族.     

负相关随机变量加权和的强定律(英文)

在本文中我们讨论了不同分布负相关随机变量加权和的强定律.在一个有限矩生成函数的条件下,一些有关负相关随机变量加权和的强定律被获得.这些结果推广了Soo HakSung[4]关于独立同分布随机变量的相应结论.我们的结果也概括了Mi Hwa Ko和Tae SungKim[7]获得的相关结论,同时使得Nili Sani H R和Bozorgnia A[9]所取得的结果更加形象.     

A Linear Risk Model

A special structure optimization model is presented which includes many of the single variable risk problems that are encountered in operational problems. A risk function is assumed which is a piece-wise linear function of some random variable whose distribution is known; one seeks the value of the decision variable which minimizes expected risk. In this paper are presented the necessary and sufficient conditions for this optimization for random variables which are either continuously or discretely distributed. The important special case of a continuous risk function is discussed; multiple risk problems with a joint constraint are analyzed; and the change in policy for a small change in the distribution of the random variable is investigated. Examples illustrate the application of the model.     

二维随机变量条件分布函数教学的思考

本文利用条件概率的定义,由随机变量分布函数的性质,给出一般情形下随机变量条件分布函数的定义,以帮助学生更好地理解随机变量的条件分布函数的概念.     

Two approximations of the present value distribution of a disability annuity

The distribution function of the present value of a cash flow can be approximated by means of a distribution function of a random variable, which is also the present value of a sequence of payments, but with a simpler structure. The corresponding random variable has the same expectation as the random variable corresponding to the original distribution function and is a stochastic upper bound of convex order. A sharper upper bound can be obtained if more information about the risk is available. In this paper, it will be shown that such an approach can be adopted for disability annuities (also known as income protection policies) in a three state model under Markov assumptions. Benefits are payable during any spell of disability whilst premiums are only due whenever the insured is healthy. The quality of the two approximations is investigated by comparing the distributions obtained with the one derived from the algorithm presented in the paper by Hesselager and Norberg [Insurance Math. Econom. 18 (1996) 35–42].     

渐近正态随机变量函数的极限分布

基于渐近正态随机变量,导出随机变量函数极限分布的两个一般性理论结果.作为应用,证明了渐近正态随机变量一系列具体函数的极限分布,其中包括泊松随机变量平方根的渐近正态性,以及随机变量部分和在正则化常数是随机变量情况下的渐近正态性.     

模糊密度随机变量的数学描述

研究了由于概率密度函数的模糊性而引起的模糊概率随机变量问题。给出了区间密度函数、模糊密度函数、模糊密度随机变量及其分布函数和模糊密度随机变量的模糊数学期望、模糊方差等基本概念及定义和计算方法,并证明了有关定理。     

Singularity of distributions of random variables given by distributions of elements of the corresponding continued fraction

The structure of the distribution of a random variable for which elements of the corresponding elementary continued fraction are independent random variables is completely studied. We prove that the distribution is pure and the absolute continuity is impossible, give a criterion of singularity, and study the properties of the spectrum. For the distribution of a random variable for which elements of the corresponding continued fraction form a uniform Markov chain, we describe the spectrum, obtain formulas for the distribution function and density, give a criterion of the Cantor property, and prove that an absolutely continuous component is absent.     

On characterizations of distributions by mean absolute deviation and variance bounds

In this paper we present a bound for the mean absolute deviation of an arbitrary real-valued function of a discrete random variable. Using this bound we characterize a mixture of two Waring (hence geometric) distributions by linearity of a function involved in the bound. A double Lomax distribution is characterized by linearity of the same function involved in the analogous bound for a continuous distribution. Finally, we characterize the Pearson system of distributions and the generalized hypergeometric distributions by a quadratic function involved in a similar bound for the variance of a function of a random variable.     

An efficient third-moment saddlepoint approximation for probabilistic uncertainty analysis and reliability evaluation of structures

This paper presents an efficient third-moment saddlepoint approximation approach for probabilistic uncertainty analysis and reliability evaluation of random structures. By constructing a concise cumulant generating function (CGF) for the state variable according to its first three statistical moments, approximate probability density function and cumulative distribution function of the state variable, which may possess any types of distribution, are obtained analytically by using saddlepoint approximation technique. A convenient generalized procedure for structural reliability analysis is then presented. In the procedure, the simplicity of general moment matching method and the accuracy of saddlepoint approximation technique are integrated effectively. The main difference of the presented method from existing moment methods is that the presented method may provide more detailed information about the distribution of the state variable. The main difference of the presented method from existing saddlepoint approximation techniques is that it does not strictly require the existence of the CGFs of input random variables. With the advantages, the presented method is more convenient and can be used for reliability evaluation of uncertain structures where the concrete probability distributions of input random variables are known or unknown. It is illustrated and examined by five representative examples that the presented method is effective and feasible.     

删失截断情形下失效率变点模型的Bayes参数估计

通过添加部分缺失寿命变量数据,得到了删失截断情形下失效率变点模型相对简单的似然函数.讨论了所添加缺失数据变量的概率分布和随机抽样方法.利用Monte Carlo EM算法对未知参数进行了迭代.结合Metropolis-Hastings算法对参数的满条件分布进行了Gibbs抽样,基于Gibbs样本对参数进行估计,详细介绍了MCMC方法的实施步骤.随机模拟试验的结果表明各参数Bayes估计的精度较高.