Probability theory

There are experiments with more than one outcome for any trial. If we do not know which outcome will result in a given trial, we define outcomes as random and we assign a number to each outcome, called the probability. We present two distinct definitions of probability  

Classical probability. Given W possible simple outcomes to an experiment or measurement, the classical probability of a simple event Ei is defined as 

If the experiment is tossing a coin, there are W = 2 possible outcomes E = heads, E2 = tails. The probability of each outcome is 

Statistical probability. If an experiment is conducted N times and an event Ei occurs n, times (n, N), the statistical probability of this event is 

The statistical probability converges to the classical probability when the number of trials is infinite. If the number of tri s is small, then the value of the statistical probability fluctuates. We show later in this chapter that the magnitude of fluctuations in the value of P(Ei) is inversely proportional to /iv. 

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Probability theory shows that tire standard deviation of a quantity v can be written as... 

We are now going to use this distribution fiinction, together with some elementary notions from mechanics and probability theory, to calculate some properties of a dilute gas in equilibrium. We will calculate tire pressure that the gas exerts on the walls of the container as well as the rate of eflfiision of particles from a very small hole in the wall of the container. As a last example, we will calculate the mean free path of a molecule between collisions with other molecules in the gas. 

There are two central rules of probability theory on which Bayesian inference is based [30] ... 

Note that the bar above y m y m this section denotes the average of y. A bar over a statement or hypothesis A in the previous section was used to denote not-A. Both of these are standard notations in statistics and probability theory, respectively. 

ET Jaynes. Probability Theory The Logic of Science, http //bayes.wustl.edu/etj/prob.html. 1999. 

The development of the probabilistic design approach, as already touched on, includes elements of probability theory and statistics. The introductory statistical methods discussed in Appendix I provide a useful background for some of the more advanced topics covered next. Wherever possible, the application of the statistical methods is done so through the use of realistic examples, and in some cases with the aid of computer software. 

Fuzzy logie was first proposed by Zadeh (1965) and is based on the eoneept of fuzzy sets. Fuzzy set theory provides a means for representing uneertainty. In general, probability theory is the primary tool for analysing uneertainty, and assumes that the... 

The central concept of fuzzy set theory is that the membership function /i, like probability theory, can have a value of between 0 and 1. In Figure 10.3, the membership function /i has a linear relationship with the x-axis, called the universe of discourse U. This produces a triangular shaped fuzzy set. 

W. Feller, "Probability Theory and Its Applications", John Wiley and Sons,... 

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. 

A successful method to obtain dynamical information from computer simulations of quantum systems has recently been proposed by Gubernatis and coworkers [167-169]. It uses concepts from probability theory and Bayesian logic to solve the analytic continuation problem in order to obtain real-time dynamical information from imaginary-time computer simulation data. The method has become known under the name maximum entropy (MaxEnt), and has a wide range of applications in other fields apart from physics. Here we review some of the main ideas of this method and an application [175] to the model fluid described in the previous section. 

Choose a site on the lattiee. That ean be done either in a systematie or a random way, but the latter method requires more eomputing time. Draw the value of the adsorption energy, from the speeified interval, aeeording to the assumed form of x( ) and assign this value to the ehosen site. (The proeedures to generate random sequenees of numbers aeeording to a given probability distribution ean be found in many textbooks on probability theory [67] and eomputer simulation methods [52].)... 

Cyrus Dcrman, Leon Glasser, and Ingram Olkin, A Guide to Probability Theory and Its Applications, Holt, New York, 1973. 

In mathematics, Laplace s name is most often associated with the Laplace transform, a technique for solving differential equations. Laplace transforms are an often-used mathematical tool of engineers and scientists. In probability theory he invented many techniques for calculating the probabilities of events, and he applied them not only to the usual problems of games but also to problems of civic interest such as population statistics, mortality, and annuities, as well as testimony and verdicts. 

The proof of this relationship is one of the triumphs of probability theory. The underlying considerations will be outlined here because they are basic to an understanding of the counting error. These considerations are most obviously applicable to radioactive systems, and it was to these that they were first applied.3... 

The first complete statement of the theory was given by A. N. Kolmogorov, Foundation of Probability Theory, Chelsea Publishing Co., New York, I960. 

For a more complete discussion Qf the material in this and the next section, see W. Feller, Probability Theory and Its Application, John Wiley and Sons, Inc., New York, 1950, and E. Parzen, Modern Probability Theory and Ite Applicatione, John Wiley and Sons, Inc., New York, 1960. 

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. 

We leave it to the reader to pursue the rigorous approach to probability theory in appropriate texts. 

Jansons, KM Phillips, CG, On the Application of Geometric Probability Theory to Polymer Networks and Suspensions, I, Journal of Colloid and Interface Science 137, 75, 1990. 

Bayesian probability theory and methods that are based on fuzzy-set theory. The principles of both theories are explained in Chapter 16 and Chapter 19, respectively. Both approaches have advantages and disadvantages for the use in expert systems and it must be emphasized that none of the methods, developed up to now are satisfactory [7,11]. 

Dwass, Meyer, Probability Theory and Applications, Benjamin Company, White Plains, New York (1970). 

In the development of probability theory, as applied to a system of particles, it is necessary to specify the distribution of particles over die various energy levels of a system. The energy levels may be clearly separated in a quantized system or approach a continuum in the classical limit. The notion of probability is introduced with the aid of the general relation... 

Although screening tests are evaluated qualitatively, as a rule, quantitative aspects of test statistics and probability theory have to be considered. In this respect, validation of qualitative analytical procedures has been included in international programs and concepts, see Trullols et al. [2004]. 

The reliability of screening methods is usually expressed in terms of probability theory. In this regard, the conditional probability, P(B A), characterizing the probability of an event B given that another event A occurs, plays an important role. 

We are also developing an improved approach, based on probability theory, for smoothing the observed data and for describing the features in orientation distributions. Since this approach relies heavily on non-linear least squares techniques, it is best done off line. 

Davis, J.M., Blumberg, L.M. (2005). Probability theory for number of mixture components resolved by n independent columns. J. Chromatogr. A 1096, 28-39. 

In this section, we take an approach that is characteristic of conventional perturbation theories, which involves an expansion of a desired quantity in a series with respect to a small parameter. To see how this works, we start with (2.8). The problem of expressing ln(exp (—tX)) as a power series is well known in probability theory and statistics. Here, we will not provide the detailed derivation of this expression, which relies on the expansions of the exponential and logarithmic functions in Taylor series. Instead, the reader is referred to the seminal paper of Zwanzig [3], or one of many books on probability theory - see for instance [7], The basic idea of the derivation consists of inserting... 

Feller, W. "An Introduction to Probability Theory and Its Applications" John Wiley New York, 1957. 

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