Mathematical Sciences: Some Problems in Simulations and Applied Probability

建立了一个基本的概率恒等式，并应用于获得新的模拟估计量，涉及(A)系统可靠性，(b)多值系统，以及(c)当数据由平稳马尔可夫链生成时，直到给定模式发生的时间的概率分布。结果表明，当通常估计量的方差为a阶且a较小时，新估计量的方差通常为a ^ 2阶。研究者将指出该估计器如何与标准方差减少技术相结合。对估计器的进一步研究，以及它的变化，以及其他应用，都被提出。研究还涉及如何使用恒等式来提供应用概率中的近似和边界。指出了一般界限和具体应用，并提出了进一步研究的建议。许多涉及偶然现象的问题太复杂，无法得到解析解。在这种情况下，概率分析和计算机模拟相结合，对感兴趣的数量进行统计估计，通常是一种有效的攻击方法。研究者指出了如何有效地利用基本概率恒等式来获得有效的模拟估计量。本研究的目的是研究其在仿真和概率分析阶段在各种应用中的实用性。研究者还指出了身份的扩展以及潜在的应用领域。* * *

Ross A basic probability identity is established and applied to obtain new simulation estimators concerning (a) system reliability, (b) a multi-valued system, and (c) the probability distribution of the time until a given pattern occurs when data is generated by a stationary Markov chain. It has been shown that the variance of this new estimator is often of the order a-squared when the usual estimator's variance is of order a and a is small. The investigator will indicate how this estimator can be combined with standard variance reduction techniques. Additional studies of the estimator, as well as on variations of it are proposed as are additional applications. The research also concerns how the identity can be used to provide approximations and bounds in applied probability. General bounds as well as specific applications are indicated and proposed for further study. Many problems involving chance phenomena are too complex for one to obtain analytical solutions. In such cases a combination of a probabilistic analysis and a computer simulation leading to statistical estimators of the quantities of interest is often a potent method of attack. The investigator has indicated how a basic probability identity can be effectively employed in obtaining efficient simulation estimators. The purpose of this research is to study its usefulness, both in the simulation and the probabilistic analysis phase, in a variety of applications. The investigator has also indicated extensions of the identity along with potential areas of applications. ***