Mathematical Sciences: Some Problems in Simulations and Applied Probability

罗斯 一个基本的概率恒等式的建立和应用，以获得新的模拟估计有关（a）系统的可靠性，（B）多值系统，和（c）的概率分布的时间，直到一个给定的模式发生时，数据产生的平稳马尔可夫链。 它已被证明，这种新的估计量的方差往往是一个平方的顺序时，通常的估计量的方差是一个订单，a是小的。 研究者将指出如何将该估计量与标准方差缩减技术相结合。 额外的研究估计，以及它的变化，提出了作为额外的应用程序。该研究还涉及如何可以使用的身份，以提供近似和应用概率的界限。 一般的界限，以及具体的应用，并提出了进一步的研究。 许多涉及偶然现象的问题太复杂了，人们很难得到解析解。 在这种情况下，概率分析和计算机模拟相结合，导致感兴趣的数量的统计估计，往往是一个强大的攻击方法。 研究者指出了如何一个基本的概率恒等式可以有效地获得有效的模拟估计。 本研究的目的是研究其实用性，无论是在模拟和概率分析阶段，在各种应用。 调查员还指出了身份的扩展沿着潜在的应用领域。***

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. ***