Probability theory random variable

ELEMENTARY DECISION THEORY, Herman Chemoff and Lincoln E. Moses. Clear introduction to statistics and statistical theory covers data processing, probability and random variables, testing hypotheses, much more. Exercises. 364pp. 5X x 8H. 65218-1 Pa. 38.95... 

Uncertainty theory is also referred to as probability theory, credibility theory, or reliability theory and includes fuzzy random theory, random fuzzy theory, double stochastic theory, double fiizzy theory, the dual rough theory, fiizzy rough theory, random rough theory, and rough stochastic theory. This section focuses on the probability theory and fiizzy set theory, including probability spaces, random variables, probability spaces, credibility measurement, fuzzy variable and its expected value operator, and so on. 

The theory is concerned with the problem of determining the probability of failure of a part which is subjected to a loading stress, L, and which has a strength, S. It is assumed that both L and S are random variables with known PDFs, represented by f S) and f L) (Disney et al., 1968). The probability of failure, and hence the reliability, can then be estimated as the area of interference between these stress and strength functions (Murty and Naikan, 1997). 

The master equation approach considers the state of a spur at a given time to be composed of N. particles of species i. While N is a random variable with given upper and lower limits, transitions between states are mediated by binary reaction rates, which may be obtained from bimolecular diffusion theory (Clifford et al, 1987a,b Green et al., 1989a,b, 1991 Pimblott et al., 1991). For a 1-radi-cal spur initially with Ng radicals, the probability PN that it will contain N radicals at time t satisfies the master equation (Clifford et al., 1982a)... 

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. 

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. 

A stochastic theory provides a simple model for chromatography.11 The term stochastic implies the presence of a random variable. The model supposes that, as a molecule travels through a column, it spends an average time Tm in the mobile phase between adsorption events. The time between desorption and the next adsorption is random, but the average time is Tm. The average time spent adsorbed to the stationary phase between one adsorption and one desorption is rs. While the molecule is adsorbed on the stationary phase, it does not move. When the molecule is in the mobile phase, it moves with the speed ux of the mobile phase. The probability that an adsorption or desorption occurs in a given time follows the Poisson distribution, which was described briefly in Problem 19-21. 

If we consider the random variable theory, this solution represents the residence time distribution for a fluid particle flowing in a trajectory, which characterizes the investigated device. When we have the probability distribution of the random variable, then we can complete more characteristics of the random variable such as the non-centred and centred moments. Relations (3.110)-(3.114) give the expressions of the moments obtained using relation (3.108) as a residence time distribution. Relation (3.114) gives the two order centred moment, which is called random variable variance ... 

At this point in this chapter, it is easy to understand that, using the methodology above, the modelling of a chemical transformation presents no important difS-culty if the chemical reaction is fitted in the general framework of the concepts of probability theory. Indeed, the discrete molecular population characterizing a chemical system can be described in terms of the joint probability of the random variables representing the groups of entities in the total population. 

For the formal theorems and proofs, it is useful if the reader is familiar with elementary information theory (see [Shan48] and [Gall68, Sections 2.2 and 2.3]). The most important notions are briefly repeated in the notation of [Gall68]. It is assumed that a common probability space is given where all the random variables are defined. Capital letters denote random variables and small letters the corresponding vadues, and terms like P(x) are abbreviations for probabilities like P(X = x). The joint random variable of X and Y is written as X, Y. The entropy of a random variable X is... 

In his treatise "The local structure of turbulence in an incompressible viscous liquid at very high Reynolds numbers , Kolmogorov [289] considered the elements of free turbulence as random variables, which are in general terms accessible to probability theory. This assumes local isotropic turbulence. Thus the probability distribution law is independent of time, since a temporally steady-state condition is present. For these conditions Kolmogorov postulated two similarity hypotheses ... 

Since distributions describing a discrete random variable may be less familiar than those routinely used for describing a continuous random variable, a presentation of basic theory is warranted. Count data, expressed as the number of occurrences during a specified time interval, often can be characterized by a discrete probability distribution known as the Poisson distribution, named after Simeon-Denis Poisson who first published it in 1838. For a Poisson-distributed random variable, Y, with mean X, the probability of exactly y events, for y = 0,1, 2,..., is given by Eq. (27.1). Representative Poisson distributions are presented for A = 1, 3, and 9 in Figure 27.3. 

The theory can be extended to reaction systems with disordered kinetics, for which the rate coefficients are random. There are two different cases (1) static disorder, for which the rate coefficients are random variables selected from certain probability laws and (2) dynamic disorder, for which the rate coefficients are random functions of time. For details see [5]. Here we give the expression of the average probability density of the lifetime for stationary systems with static disorder ... 

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. 

Hence, by assuming nearly Gaussian distributed random variables we are able to compute the exercise probabilities Tip [.ST] directly by performing tbe lEE approach instead of miming a generalized Edgeworth series expansion. Overall, the lEE approach (4.2) can be seen as an equivalent to tbe generalized EE tecbnique, especially adapted to compute tbe cdf s used in finance theory. 

Models that seek to value options or describe a yield curve also describe the dynamics of asset price changes. The same process is said to apply to changes in share prices, bond prices, interest rates and exchange rates. The process by which prices and interest rates evolve over time is known as a stochastic process, and this is a fundamental concept in finance theory. Essentially, a stochastic process is a time series of random variables. Generally, the random variables in a stochastic process are related in a non-random manner, and so therefore we can capture them in a probability density function. A good introduction is given in Neftci (1996), and following his approach we very briefly summarise the main features here. 

Theoretical probabilistic safety assessment relies on identifying the structure parameters and the physical model which governs the failure (Elegbede 2005). This physical model is generally described by a function named limit state function in structural reliability theory (Madsen et al. 2006). Since the physical parameters of the structures are generally covered with uncertainties, there are generally modeled by random variables. Thus, to assess failure probability of... 

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