Probability, statistics and

In Chapters 2 and 3 we studied the formulation of models from chemical and physical principles and the solution of models by numerical methods. In this chapter we begin the study of statistical methods and their role in model development. 

Statistics is the science of efficient planning and use of observations with the aid of probability theory. It began with investigations in astronomy later it came into wide use throughout science and engineering. The following areas of statistics are considered in this book  

Inference Estimation of process parameters and states from data. See Chapters 5-7 and Appendix C. 

Criticism Analysis of residuals (departures of data from fitted models). This topic is included in Chapters 6 and 7 and Appendix C. 

Design of Experiments Construction of efficient test patterns for these activities. Factorial designs are well treated by Box, Hunter, and Hunter (1978). Sequential procedures for experimental design are presented in Chapters 6 and 7 and Appendix C, and in the references cited there. 

1 Would a base 4 or 5 system of numbering be beneficial in devising mathematical methods for handling problems dealing with information transfer in DNA and RNA research  

2 What is the total number of configurations possible in a polymer chain composed of 150 monomers each of which can take one of four conformations  

6 Graphically show the distribution function of a uniformly distributed random variable. 

7 You have 20 amino acids available. How many pentapeptides can be made if an amino acid is not replaced after being used  

8 You have 20 adenosines, 50 thymidines, 15 guanosines, and 35 cytosines. Pick 15 sequential nucleosides. What is the probability of getting 5 thymidine,. 5 adenosine,. 5 cystosine and 1 guanosine  

We suppose that the reader is familiar with the basic concepts of probability and statistics. It should be noted that although applied statistics is a very pragmatical discipline, its theoretical base (the probability theory) is surprisingly abstract. We shall not, however, start from the fundamental concepts of probability space and probability measure. Rather, we will rely upon the reader s intuition concerning the notion of probability itself. In this book, we are interested in probability on vector spaces (spaces of several numeric variables). At a slightly advanced level, let us begin with the concept of a random vector variable. The reader is recommended to peruse Appendix B, in particular Sections B.9-11. 

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The integral of the Gaussian distribution function does not exist in closed form over an arbitrary interval, but it is a simple matter to calculate the value of p(z) for any value of z, hence numerical integration is appropriate. Like the test function, f x) = 100 — x, the accepted value (Young, 1962) of the definite integral (1-23) is approached rapidly by Simpson s rule. We have obtained four-place accuracy or better at millisecond run time. For many applications in applied probability and statistics, four significant figures are more than can be supported by the data. 

H. Wold, ia J. Gani, ed.. Perspectives in Probability and Statistics Academic Press, New York, 1975. 

R von Mises. Mathematical Theory of Probability and Statistics. New York Academic Press, 1964. 

Irwin Miller and Jolm E. Freund, Probability and Statistics for Engineers, 3" ed., Prentice-Hall, Englewood Cliffs, NJ, 1985. 

Probability Theory.—To pursue our study of methods of operations research, a brief, although incomplete, and somewhat abstract, presentation of ideas from probability theory will be given. In part it shows that mathematical abstraction and rigor are also in the nature of operations research. Illustrations of this topic will be given in later sections. We then give a longer discussion of maximization and minimization methods and in turn illustrate the ideas in subsequent sections. Probability and statistics and optimization methods are two major sources of operations research tools. 

Moran, P. A. P., The Theory of Storage, Methuen s Monographs on Applied Probability and Statistics, John Wiley and Sons, Inc., New York, 1959. 

H. Wold, Soft modelling by latent variables the non-linear iterative partial least squares (NIPALS) algorithm. In Perspectives in Probability and Statistics, J. Gani (Ed.). Academic Press, London, 1975, pp. 117-142. 

Plackett, R.L., "Studies in the History of Probability and Statistic. XXIX The Discovery of the method of least squares", Biometrika, 59 (2), 239-251 (1972). 

MacQueen, J., Some methods for classification and analysis of multivariate data, Proc. 5th Berkeley Symp. on Probability and Statistics, Berkeley, CA (1967). 

In the remainder of this chapter, an overview of the CRE and FM approaches to turbulent reacting flows is provided. Because the description of turbulent flows and turbulent mixing makes liberal use of ideas from probability and statistical theory, the reader may wish to review the appropriate appendices in Pope (2000) before starting on Chapter 2. Further guidance on how to navigate the material in Chapters 2-7 is provided in Section 1.5. 

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. 

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