The theory of probability

A stochastic system is one whose behavior is not purely deterministic and predictable, but rather has (assumed inherent) randomness. The theory of probability provides a mathematical fi amework for understanding and modeling such systems. Rather than provide abstract definitions, we introduce the subject through an example modeling the distribution of polymer chain lengths in condensation polymerization. 

Chain length distribution in linear condensation polymers joint and conditional probabilities 

we consider the linear case with oii = P2 = 2. Initially, we have equal concentrations of the two monomers, [Mi]q = [M2]o- As each monomer has two end groups, die initial acid and base concentrations are also equal, [A]q = [B]o = 2[Mi]o. As the acid and base groups react, they form longer and longer chains. How does die distribution of chain lengths vary as a function of conversion  

We define the conversions of the acid and base groups respectively as 

Because [A]q = [B]q and the reaction consumes equal numbers of acid and base end groups, px = p = P- 0 p , let [P ] be the concentration of chains that comprise n monomer units, i.e., that have a chain length or degree of polymerization equal to n. Each of the monomers has a chain length of 1 thus, at p = 0, [Pi] = 2[MiJo = [A]q and all other [P i] = 0. 

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At this point, it is appropriate to mention an elementary concept from the theory of probability. If there are n possible numerical outcomes associated with a particular process, the average value ( ) can be calculated by simnning up all of the outcomes, each weighted by its corresponding probability... 

The study or errors in a computation is related to the theory of probability. In what follows a relation for the error will be given in certain instances. 

Gnedenko B. V. The Theory of Probability (Chelsea, New York) (1962). [Translated from Russian Teoria veroyatnostey Moscow GITTL, (1954)]. 

Statistical inference. The broad problem of statistical inference is to provide measures of the uncertainty of conclusions drawn from experimental data. This area uses the theory of probability, enabling scientists to assess the reliability of their conclusions in terms of probability statements. 

In the theory of probability the term correlation is normally applied to two random variables, in which case correlation means that the average of the product of two random variables X and Y is the product of their averages, i.e., X-Y)=(,XXY). Two independent random variables are necessarily uncorrelated. The reverse is usually not true. However, when the term correlation applies to events rather than to random variables, it becomes equivalent to dependence between the events. 

The definition of correlation functions in this book differs from the definition of the correlation coefficient in the theory of probability. The difference is essentially in the normalization, i.e., whereas g(, ) can be any positive number 0 S g the correlation coefficient varies within [-1,1]. We have chosen the definition of correlation as in Eq. (1.5.19) or (1.5.20) to conform with the definition used in the theory of liquids and solutions. 

In its simplest form, information theory concerns itself with the concrete verifiable statements (the events ) of the theory of probability. If a is such a statement and if all that is known initially concerning its truth is its probability P(a), then the quantity of information brought in by the subsequent knowledge that it is in fact true is - c log P(a). Here the positive constant c depends on the unit of information we take c = 1 henceforth. 

To understand the nature of the uncertainty in selecting a sample for analysis, consider a random mixture of two kinds of solid particles. The theory of probability allows us to state the likelihood that a randomly drawn sample has the same composition as the bulk sample. It may surprise you to learn how large a sample is required for accurate sampling.9... 

Excursus. The theory of probability is nothing but transforming variables. It comprises a collection of techniques for transforming an a priori given distribution into another one, called the a posteriori distribution. Any application to real phenomena consists of the following three steps. 

Then Y is an element of a non-standard sequence if Y e R and M e N. We say that YM converges to Y if Y - st Y for all infinite integers M. Continuity and differentiability can be similarly defined, for example ff(a) = st (f(a+h) - f(a) /h for all infinitesimals h. The Dirac delta function can be defined pointwise and new approaches to the theory of probability and stochastic differential equations are opened up. But these are applications and paradoxically I have been describing the abstraction that is the non-standard world in very concrete terms. This is no place to go back and try to present it abstractly, but it stands as an example of a recent step forward in the path of abstraction that began two and a half millenia ago (see also (23,24)). 

A. N. Kolmogorov, Foundations of the Theory of Probability (N. Morrison, transl.), Chelsea, New York, 1950. The original, Grundbegriffe der Wahrscheinlichkeitsrechnung, was published in Berlin in 1933. 

One can see that the truth values in fuzzy logic strongly resemble the stochastic values from the theory of probabilities. However, methods based on the use of statistics are not considered fuzzy by the orthodox fuzzy theory protagonists. Instead of using probability values, fuzzy theory works with possibility values. It is argued that both values are substantially different and that the latter have to be evaluated by methods other than statistical. Our understanding, however, is that at a very fundamental level, both values have essentially the same nature. 

Pascal, Blaise—Blaise Pascal (1623-1662), a well known mathematician, was a founder of the theory of probability. The combinatorial triangle was given his name when he published a paper compiling the previous work done by the Hindus, Chinese, and Creeks. 

Of course, the foregoing calculations deal with an extreme case of non-uniform distribution. Nevertheless, it is possible to show by means of the theory of probability that the chances of any appreciable spontaneous fluctuation from a uniform distribution of the gas throughout the whole available space is so extremely small that it is unlikely to be observed in millions of years, provided the system contains an appreciable number of molecules. It is possible to state, therefore, that the probability of the virtually uniform distribution of a considerable number of molecules in the space available is very large. 

The above-mentioned equation for n in fact is only valid for isocratic separations and if the peaks are symmetric the peak capacity is larger with gradient separations. Tailing decreases the peak capacity of a column. In real separations the theoretical plate number is not constant over the full k range. However, it is even more important to realize that a hypothetical parameter is discussed here. It is necessary to deal with peaks that are statistically distributed over the accessible time range. The theory of probabilities allows us to proceed from ideal to near-real separations. Unfortunately, the results are discouraging. 

The treatment of statistics is focused on explicit applications of both linear and nonlinear least-squares methods, rather than on the alphabet soup (F, Q, R, T, etc.) of available tests. However, within that rather narrow framework, many practical aspects of error analysis and curve fitting are considered. They are chosen to illustrate the now almost two centuries old dictum of de Laplace that the theory of probability is merely common sense confirmed by calculation. 

H. Reichenbach, The Theory of Probability. Univ. California Press, Berkeley, 1949. 

All these integrals are of considerable importance in the kinetic theory of gases, and in the theory of probability. In the former we shall meet integrals like... 

Nearly every inference we make with respect to any future event is more or less doubtful. If the circumstances are favourable, a forecast may be made with a greater degree of confidence than if the conditions are not so disposed. A prediction made in ignorance of the determining conditions is obviously less trustworthy than one based upon a more extensive knowledge. If a sportsman missed his bird more frequently than he hit, we could safely infer that in any future shot he would be more likely to miss than to hit. In the absence of any conventional standard of comparison, we could convey no idea of the degree of the correctness of our judgment. The theory of probability seeks to determine the amount of reason which we may have to expect any event when we have not sufficient data to determine with certainty whether it will occur or not and when the data will admit of the application of mathematical methods. 

The theory of probability does not pretend to furnish an infallible criterion for the discrimination of an accidental coincidence from the result of a determining cause. Certain conditions must be satisfied before any reliance can be placed upon its dictum. For example, a sufficiently large number of cases must be available. Moreover, the theory is applied irrespective of any knowledge to be derived from other sources which may or may not furnish corroborative evidence. Thus KirchhofFs conclusion as to the probable existence of iron in the sun was considerably strengthened by the apparent relation between the brightness of the coincident lines in the two spectra. 

For details of the calculations, the reader must consult the original memoirs. Most of the calculations are based upon the analysis in Laplace s old but standard Theorie (l.c.). An excellent r sumk of this latter work will be found in the Encyclopedia Metro-politana. The more fruitful applications of the theory of probability to natural processes have been in connection with the kinetic theory of gases and the law relating to errors of observation. 

As a matter of fact the theory of probability is of little or no importance, when the constant, or systematic errors are greater than the accidental errors. Still further, this use of the probable error cannot be justified, even when the different series of experiments are only affected with accidental errors, because the probable error only shows how uniformly an experimenter has... 

The above method of expressing results is not mathematically altogether correct. The average number of B. coli per cc. as thus estimated is not precisely the most probable number calculated by application of the theory of probability. To apply this theory to a correct mathematical solution of any considerable series of results involves, however, mathematical calculations so complex as to be impracticable of application in general practice. The simpler method given is therefore considered preferable, eince it is easily applied, and tiie results so expressed are readily comprehensible. 

Arley, N., and Buck, K. R., Introduction to the Theory of Probability and Statistics, Wiley, New York, 1950. 

This is as far as the theory of probability can carry us. We must now introduce a model from which the conditional probabilities P(A B C) can be calculated. 

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