Random phenomena

In the history of the development of mathematics, one important branch was the study of the behavior of randomness. Initially, there were no highfalutin ideas of making science out of what appeared to be disorder rather, the investigations of random phenomena that lead to what we now know as the science of Statistics began as studies of the behavior of the random phenomena that existed in the somewhat more prosaic context of gambling. It was not until much later that the recognition came that the same random phenomena that affected, say, dice, also affected the values obtained when physical measurements were made. 

By the time this realization arose, it was well recognized that random phenomena were describable only by probabilistic statements by definition it is not possible to state a priori what the outcome of any given random event will be. 

The variation superimposed by any random phenomena affecting the data. 

The remaining errors in the data are usually described as random, their properties ultimately attributable to the nature of our physical world. Random errors do not lend themselves easily to quantitative correction. However, certain aspects of random error exhibit a consistency of behavior in repeated trials under the same experimental conditions, which allows more probable values of the data elements to be obtained by averaging processes. The behavior of random phenomena is common to all experimental data and has given rise to the well-known branch of mathematical analysis known as statistics. Statistical quantities, unfortunately, cannot be assigned definite values. They can only be discussed in terms of probabilities. Because (random) uncertainties exist in all experimentally measured quantities, a restoration with all the possible constraints applied cannot yield an exact solution. The best that may be obtained in practice is the solution that is most probable. Actually, whether an error is classified as systematic or random depends on the extent of our knowledge of the data and the influences on them. All unaccounted errors are generally classified as part of the random component. Further knowledge determines many errors to be systematic that were previously classified as random. 

Percolation theory provides a well-defined model applicable to a wide variety of spatially random phenomena, both macroscopic and microscopic (Table X). The characteristic length scales for these phenomena... 

Unfortunately, it is impossible to design an experiment that will totally disprove a theory based on random phenomena. Various outcomes may occur, some of which may be unlikely but not impossible. Thus Popper s falsifiability condition does not apply. The statistical method advocated by Fisher (1956) attempts to overcome this problem by substituting unlikely for impossible but otherwise follows the principles of the scientific method. With this substitution, Fisher and others proposed conceptual structures for testing theories and scientific hypotheses under conditions of uncertainty that are analogous to the scientific method. However, these approaches, although being very useful in practice, have raised a host of conceptual issues that are the subject of ongoing debates. 

Nuclear decay, the emission of radiation from a decaying atom, and the detection of emitted radiation by a detector are inherently random phenomena. Their occurrences cannot be predicted with certainty, even in principle, although they can be described probabilistically. The randomness of these processes causes the result of a radiation-counting measurement to vary when the measurement is repeated and thus leads to an uncertainty in the result, called the counting uncertainty. ... 

The scientific method runs into difficulties when applied to random phenomena. A random phenomenon is one where the outcome cannot be predicted with certainty from the experimental conditions. One cannot guarantee the repeat of a coin toss, no matter how hard one tries to keep the conditions constant. Neither can one expect a drug to produce an identical effect in the same patient under identical conditions on separate occasions. Such phenomena can be described probabilistically. That is, one can assign numerical values describing the likelihood, or probability, of the possible outcomes. 

One might define the function of applied statistics as the ait and science of collecting euid processing data in order to meike inferences aixmt the parameters of one or more populations associated with random phenomena. These inferences are made in such a way that the conclusions reached are consistent and unbiased. When properly applied and executed, statistical procedures depend entirely on specific methodologies, definitions, and parameters required by the statistical test chosen. 

Each (and every) random variable has a unique probability distribution. For the most part statisticians deal with the theory of these distributions. Engineers, on the other hand, are mostly interested in finding factual knowledge about certain random phenomena, by way of probability distributions of the variables directly involved, or other related variables. 

In basic statistics we learn that probability density functions can be defined by certain constants called distribution parameters. These parameters in turn can be used to characterize random variables through measures of location, shape, and variability of random phenomena. The most important parameters are the mean p and the variance The parameter /r is a measure of the center of the distribution (an analogy is the center of gravity of a mass) while is a measme of its spread or range (an analogy being the moment of inertia of a mass). Hence, when we speak of the mean and the variance of a random variable, we refer to two statistical parameters (constants) that greatly characterize or influence the probabilistic behavior of the random variable. The mean or expected value of a random variable x is defined as... 

Thus, it is not difficult to see why the field of probabihty and statistics is a discipline within itself, nor is it difficult to see why almost every discipline in existence needs a working knowledge of statistics. Random phenomena (variables) exist in all phases of activity. 

All statistical methods have, as a basis, a sample of n observations on the random phenomena of interest. Such methods require that the random sample (of size n) be representative of the outcomes that could occur. Because of this, much is said about making sure that the sample is a random sample. Many statistical calculations become invalid when the data used are not representative. 

The Mars Polar Lander loss is a component interaction accident. Such acddents arise in the interactions among system components (electromechanical, digital, human, and social) rather than in the failure of individual components. In contrast, the other main type of accident, a component failure accident, results from component failures, including the possibility of multiple and cascading failures. In component failure accidents, the failures are usually treated as random phenomena. In component interaction accidents, there may be no failures and the system design errors giving rise to unsafe behavior are not random events. 

So, if the imperfections of E t) he in that ballpark, the resolution would be near-independent of the ion current the postulated redistribution of ions in the gap (4.3.5), though continuous rather than periodic, may be caused by noise on the FAIMS waveform. However, other randomizing phenomena may be as or more important (4.3.5) and the origins of peak smearing in FAIMS remain uncertain. 

Although the Ordinary Petri Nets are a powerfirl modelling tool, many extensions and abbreviations exist. So for example Timed Petri Nets allow the incorporation of time attributes into the model. For modelling of various random phenomena Stochastic Petri Nets can be used. Colored Petri Nets (Jensen, K. 1997) allow the token differentiation due to the association of a color (value) with each token. They also give the possibility to model a system in which cooperate continuous-time variables and discrete events, which can occur on a stochastic basis. CPN allow thus to make the model more concise than the ordinary Petri Nets and also to include temporal and stochastic proprieties. 

KEISHIRO SHIRAHAMA is Professor of physical chemistry of St a University, Saga, Japan. He received a doctor of science from Kyushu University. His academic interest is directed to amphiphilic molecular assemblies such as surfactant-polymer complex, mixed micelle, and vesicle as well as random phenomena in physicochemical systems. 

Fuzzy random variables describe the fuzzy random phenomena via mathematical tools which were originally proposed by Kwakemaak [34, 35] and developed by Puri and Ralescu [36, 37], Kruse and Meyer [38], Liu and Liu [39]. 

Random Phenomena, Fundamentals of Probability and Statistics for Engineers, Babutnde A. Ogunnaike, CRC Press, 2010... 

Ogunnaike BA (2010) Random phenomena fundamentals of probability and statistics for caigi-neers. CRC Press, Boca Rahm... 

In Section F.l, we considered the probability of one or more events occurring. The same probability concepts are also applicable for random variables such as temperatures or chemical compositions. For example, the product composition of a process could exhibit random fluctuations for several reasons, including feed disturbances and measurement errors. A temperature measurement could exhibit random variations due to turbulence near the sensor. Probability analysis can provide useful characterizations of such random phenomena. 

In Taylor s analysis of turbulent flow the velocity field is considered random, in the sense that the variable does not have a unique value (i.e., the same value every time an experiment is repeated under the same set of conditions). Random functions of time in specific literature are also often called random processes. However, this does not mean that turbulence is a random phenomena. 

Denker M, Woyczynski WA (1998) Introductory statistics and random phenomena. Birkhaiiser, Boston/Basel/ Berlin... 

---