Probability Theory and Statistical Analysis

In the practice of modem system safety analysis, the system safety engineer attempts to provide a sufficient level of information to organizational management so that informed decisions may be made regarding hazard risk acceptance or rejection. In the safety and health arena, the provision of such choices often requires ample substantiation in order to justify decisions to accept a hazard risk. The system safety practitioner can utilize a wide variety of techniques and methods to determine risk levels and, through preestablished acceptance criteria, make recommendations to management. These analytical tools serve to qualify the risk in relation to some existing level and/or standard of operation. Some of the more common of these tools are discussed in detail in Part II of this text. When acmal failure rate data are known or can be determined or deduced, the system safety effort can take the analysis process further and actually quantify the risk of hazard in terms of these known or expected failure rates. 

While probability theory examines the likelihood of a specific failure event given a single opportunity for occurrence, statistics focuses on the number of times that a failure event will occur given many opportunities. 

Through the use of basic probability theory and statistical analysis, the system safety function can actually assign expected values to certain hazards and/or failures to determine the likelihood of their occurrence. The availability of such quantifiable information further enhances the management decisionmaking process and justifies the existence of the system safety effort within the organization. 

Basic Guide to System Safety, by Jeffrey W. Vincoli Copyright 2006 John Wiley Sons, Inc. 

This chapter presents the fundamental principles of probabiUty theory and briefly examines the use of statistical analysis in the practice of system safety. The information discussed here should provide the reader with a very basic understanding of these concepts, which, by some accounts, is essential to the overall understanding of the system safety discipline. It should be noted that it is not within the scope of this Basic Guide to System Safety to provide aU there is to know regarding probability theory and statistical analysis. However, a certain level of understanding is essential and will therefore be discussed here. 

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Therefore, Part I of this text focused primarily on the development of system safety, its military connections, the importance of including system safety requirements in contract acquisitions, the criticality of obtaining management commitment in support of the system safety effort, the process of risk analysis and assessment, probability theory and statistical analysis as they relate to system safety, and— perhaps of most value— how the fundamental principles of system safety are closely related to those of occupational safety and health management. 

The KS limits are certainly a standard idea in probability theory and have been used in traditional risk analyses, for instance as a way to express the reliability of the results of a simulation. However, it has not heretofore been possible to use KS limits to characterize the statistical reliability of the inputs. There has been no way to propagate KS limits through calculations. Probability bounds analysis allows us to do this... 

A. S. Goldman and H. D. Lewis, Particle size analysis theory and statistical methods . Chapter 1 in Handbook of Powder Science and Technology (eds. M. E. Fayed and L. Otten), Van Nostrand Reinhold Co., New York, USA, 1983, pp. 1-30. M. G. Kendall and P. A. P. Moran, Geometrical Probability y Griffin, London, UK, 1963. 

Sect. 1.3. Use your basic reference books on mathematics, statistics, and vector analysis to support the concepts and derivations developed. Some examples are (1960 and later) International Dictionary of Applied Mathematics. Van Nostrand, Princeton. Peller W (1950 and later) An Introduction to Probability Theory and Its Applications. Wiley, New York. 

We now consider probability theory, and its applications in stochastic simulation. First, we define some basic probabihstic concepts, and demonstrate how they may be used to model physical phenomena. Next, we derive some important probability distributions, in particular, the Gaussian (normal) and Poisson distributions. Following this is a treatment of stochastic calculus, with a particular focus upon Brownian dynamics. Monte Carlo methods are then presented, with apphcations in statistical physics, integration, and global minimization (simulated annealing). Finally, genetic optimization is discussed. This chapter serves as a prelude to the discussion of statistics and parameter estimation, in which the Monte Carlo method will prove highly usefiil in Bayesian analysis. 

Statistical techniques can be used for a variety of reasons, from sampling product on receipt to market analysis. Any technique that uses statistical theory to reveal information is a statistical technique, but not all applications of statistics are governed by the requirements of this part of the standard. Techniques such as Pareto Analysis and cause and effect diagrams are regarded as statistical techniques in ISO 9000-2 and although numerical data is used, there is no probability theory involved. These techniques are used for problem solving, not for making product acceptance decisions. 

In most natural situations, physical and chemical parameters are not defined by a unique deterministic value. Due to our limited comprehension of the natural processes and imperfect analytical procedures (notwithstanding the interaction of the measurement itself with the process investigated), measurements of concentrations, isotopic ratios and other geochemical parameters must be considered as samples taken from an infinite reservoir or population of attainable values. Defining random variables in a rigorous way would require a rather lengthy development of probability spaces and the measure theory which is beyond the scope of this book. For that purpose, the reader is referred to any of the many excellent standard textbooks on probability and statistics (e.g., Hamilton, 1964 Hoel et al., 1971 Lloyd, 1980 Papoulis, 1984 Dudewicz and Mishra, 1988). For most practical purposes, the statistical analysis of geochemical parameters will be restricted to the field of continuous random variables. 

Instead of determining 4, in equation (8.33), we must determine r in equation (8.34). Although the difference between Higbie s penetration theory and Danckwerts surface renewal theory is not great, the fact that a statistical renewal period would have a similar result to a fixed renewal period brought much credibility to Higbie s penetration theory. Equation (8.34) is probably the most used to date, where r is a quantity that must be determined from the analysis of experimental data. 

In view of this difficulty, verifiable statements about measured physical quantities cannot be made with unlimited exactness, but always imply an uncertainty. Nevertheless, from a large number of individual measurements one can arrive by statistical analysis at values that have a high probability of being correct and this probability can also be calculated. Properly speaking, then, any distance from the nucleus is possible for the electron however, some distances are more probable than others and there is also a most probable distance. The discovery of Heisenberg thus forces us into devising a theory that not only makes pertinent predictions but at the same time qualifies these predictions by stating a probability that the expectation is observed. 

We emphasize that the question of stability of a CA under small random perturbations is in itself an important unsolved problem in the theory of fluctuations [92-94] and the difficulties in solving it are similar to those mentioned above. Thus it is unclear at first glance how an analogy between these two unsolved problems could be of any help. However, as already noted above, the new method for statistical analysis of fluctuational trajectories [60,62,95,112] based on the prehistory probability distribution allows direct experimental insight into the almost deterministic dynamics of fluctuations in the limit of small noise intensity. Using this techique, it turns out to be possible to verify experimentally the existence of a unique solution, to identify the boundary condition on a CA, and to find an accurate approximation of the optimal control function. 

A rational deduction of elemental abundance from solar and stellar spectra had to be based on quantum theory, and the necessary foundation was laid with the Indian physicist Meghnad Saha s theory of 1920. Saha, who as part of his postdoctoral work had stayed with Nernst in Berlin, combined Bohr s quantum theory of atoms with statistical thermodynamics and chemical equilibrium theory. Making an analogy between the thermal dissociation of molecules and the ionization of atoms, he carried the van t Hoff-Nernst theory of reaction-isochores over from the laboratory to the stars. Although his work clearly belonged to astrophysics, and not chemistry, it relied heavily on theoretical methods introduced by and associated with physical chemistry. This influence from physical chemistry, and probably from his stay with Nernst, is clear from his 1920 paper where he described ionization as a sort of chemical reaction, in which we have to substitute ionization for chemical decomposition. [81] The influence was even more evident in a second paper of 1922 where he extended his analysis. [82]... 

Inference and criticism are complementary facets of statistical analysis. Both are rooted in probability theory, but they address different questions about a model. 

An alternative interpretation of the Maxwell-Boltzmann speed distribution is helpful in statistical analysis of the experiment. Experimentally, the probability that a molecule selected from the gas will have speed in the range Au is defined as the fraction AN/N discussed earlier. Because AN/N is equal to f u) Au, we interpret this product as the probability predicted from theory that any molecule selected from the gas will have speed between u and u + Au. In this way we think of the Maxwell-Boltzmann speed distribntion f(u) as a probability distribution. It is necessary to restrict Au to very small ranges compared with u to make sure the probability distribution is a continuous function of u. An elementary introdnction to probability distributions and their applications is given in Appendix C.6. We suggest you review that material now. 

We understand that chemistry is not statistics, and that books about data analysis for chemists might miss the larger statistical points. Here are a couple of books that might describe this greater picture in a way that chemists might understand. The text by Wild is more dense but does relate statistics to the underlying probability theory. 

Also using chemical space as a framework, Agrafiotis [118] presented a very fast method for diversity analysis on the basis of simple assumptions, statistical sampling of outcomes, and principles of probability theory. This method presumes that the optimal coverage of a chemical space is that of uniform coverage. The central limit theorem of probability theory... 

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