概率论思维及其智力品质的培养

概率论思维是人脑和概率论研究对象交互作用并按照一般思维规律认识概率论内容的内在理性活动.它具有随机性、概括性、问题性、辐射性、指向性和创造性.提高概率论思维的效率及质量,必须从构筑知识平台,加强应用训练及强化批判意识等方面全面注意概率论思维智力品质的培养.  

《概率论与数理统计》课程教学的探索与实践

结合多年的《概率论与数理统计》课程的教学实际,对该课程的优化教学过程,分层次教学和完善课程的考核评价等方面进行了一些探索与实践,以激发学生的学习兴趣与热情,培养学生应用概率统计方法解决问题的能力.  

正态分布两发现过程的数学文化比较

正态分布是概率论中最重要的分布,它分别由棣莫弗和高斯各自独立导出,其发现极大的促进了概率论和数理统计的发展.对正态分布两推导过程所蕴舍的概率思想及创新点进行了详细分析.另外,还比较了其不同的数学文化背景.  

A remark on probability spaces

The object of this paper is to exhibit some aspects of Kolmogorov's axiomatic theory of probability related to its phenomenological interpretation.J. P. Gabriel has informed the editor that P. Milasevic was killed in a mountain accident on October 27, 1989.  

Normalized convergence in stochastic optimization

A new concept of (normalized) convergence of random variables is introduced. This convergence is preserved under Lipschitz transformations, follows from convergence in mean and itself implies convergence in probability. If a sequence of random variables satisfies a limit theorem then it is a normalized convergent sequence. The introduced concept is applied to the convergence rate study of a statistical approach in stochastic optimization.  

The average quality of greedy-algorithms for the Subset-Sum-Maximization Problem

This paper deals with the quality of approximative solutions for the Subset-Sum-Maximization-Problem maximize $$\sum\limits_{i = l}^n {a_i x_i } $$ subject to $$\sum\limits_{i = l}^n {a_i x_i } \leqslant b$$ wherea _l,...,a_n,bεR⁺ andx _l,...x_nε{0,1}. produced by certain heuristics of a Greedy-type. Every heuristic under consideration realizes a feasible solution (x ₁, ..., x_n) whose objective value is less or equal the optimal value, which is of course not greater thanb. We use the gap between capacityb and realized value as an upper bound for the error made by the heuristic and as a criterion for quality. Under the stochastic model:a ₁, ..., a_n, b independent,a ₁...,a_n uniformly distributed on [0, 1], b uniformly distributed on [0,n] we derive the gap-distributions and the expected size of the gaps. The analyzed algorithms include four algorithms which can be done in linear time and four heuristics which require sorting, which means that they are done inO(nlnn) time.  

Existence of an optimal control for stochastic systems governed by ito equations

We give existence theorems for stochastic control problems with a lower semicontinuous cost functional and governed by Ito equations. We prove that two formulations of the fundamental problem are equivalent, one involving nonanticipative controls and the other involving (measurable) feedback controls. We then use the concept ofconvergence in distribution to prove existence for the first problem, and hence for the second as well. While our work has certain similarities with a paper of Kushner, our techniques are different and lead to more general results.  

Stochastic systems in Riemannian manifolds

A formulation of stochastic systems in a Riemannian manifold is given by stochastic differential equations in the tangent bundle of the manifold. Brownian motion is constructed in a compact Riemannian manifold as well as the horizontal lift of this process to the bundle of orthonormal frames. The solution of some stochastic differential equations in the tangent bundle of the manifold is defined by the transformation of the measure for the manifold-valued Brownian motion by a suitable Radon-Nikodym derivative. Real-valued stochastic integrals are defined for this Brownian motion using parallelism along the Brownian paths. A stochastic control problem is formulated and solved for these stochastic systems where a suitable convexity condition is assumed.This research was supported by NSF Grants Nos. GK-32136, ENG-75-06562, and MCS-76-01695.The author wishes to thank D. Gromoll, J. Simons, and J. Thorpe for some helpful conversations on differential geometry.  

A stopping rule for optimization using randomized search procedures

In many nonconvex programming problems, it is possible to locate local optima, but the global optimum may be difficult to determine. In such cases, a search procedure is often used, with random starting solutions, to find alternate local optima. This search can be terminated by a stopping rule, based upon Bayesian revised probability distributions, which determines the optimal number of iterations. The application of this rule to a resource allocation problem in project scheduling is illustrated.This work was supported in part by grants from the National Science Foundation and the Rochester Gas and Electric Corporation to the Massachusetts Institute of Technology.  

概率论与数理统计专业研究生教学改革

本文通过对概率论与数理统计专业研究生教育中存在问题的分析，提出了概率i＇e-~数理统计专业研究生教学改革的几点建议．