scholarly journals Statistical maps: A categorical approach

2007 ◽  
Vol 57 (1) ◽  
Author(s):  
Roman Frič
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Measurable Space ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Categorical Approach ◽  
Borel Sets ◽  
Fuzzy Probability Theory ◽  
Fuzzy Random

AbstractIn probability theory, each random variable f can be viewed as channel through which the probability p of the original probability space is transported to the distribution p f, a probability measure on the real Borel sets. In the realm of fuzzy probability theory, fuzzy probability measures (equivalently states) are transported via statistical maps (equivalently, fuzzy random variables, operational random variables, Markov kernels, observables). We deal with categorical aspects of the transportation of (fuzzy) probability measures on one measurable space into probability measures on another measurable spaces. A key role is played by D-posets (equivalently effect algebras) of fuzzy sets.

scholarly journals Real functions and the extension of generalized probability measures II

2015 ◽  
Vol 62 (1) ◽  
pp. 191-204
Author(s):  
Jana Havlíčková
Keyword(s):  
Probability Theory ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Random Events ◽  
Real Functions ◽  
Generalized Probability ◽  
Fuzzy Probability Theory ◽  
Fuzzy Random

Abstract We continue our study of the extensions of generalized probability measures. First, we describe some extensions of generalized random events (represented by classes of functions with values in [0,1]) to which generalized probability measures can be extended. Second, we study products of domains of probability and describe states on such products. Third, we show that the events in IF-probability, introduced by B. Riečan, form a suitable category isomorphic to a subcategory of the category of fuzzy random events. Consequently, IF-probability can be interpreted within fuzzy probability theory. We put forward some problems related to the extensions of probability domains and hint some applications.

scholarly journals On Distribution Characteristics of a Fuzzy Random Variable

2018 ◽  
Vol 47 (2) ◽  
pp. 53-67 ◽  
Author(s):  
Jalal Chachi
Keyword(s):  
Probability Theory ◽  
Distribution Function ◽  
Cumulative Distribution Function ◽  
Random Variables ◽  
Fuzzy Random Variable ◽  
Random Variable ◽  
Cumulative Distribution ◽  
Distribution Characteristics ◽  
Fuzzy Random Variables ◽  
Fuzzy Random

In this paper, rst a new notion of fuzzy random variables is introduced. Then, usingclassical techniques in Probability Theory, some aspects and results associated to a randomvariable (including expectation, variance, covariance, correlation coecient, etc.) will beextended to this new environment. Furthermore, within this framework, we can use thetools of general Probability Theory to dene fuzzy cumulative distribution function of afuzzy random variable.

Statistical maps and generalized random walks

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Measure ◽  
Random Walks ◽  
Measurable Space ◽  
Probability Measures ◽  
Statistical Map ◽  
Probability Spaces ◽  
Special Matrix ◽  
Fuzzy Probability Theory ◽  
Fuzzy Random ◽  
Joint Probabilities

AbstractWe continue our study of statistical maps (equivalently, fuzzy random variables in the sense of Gudder and Bugajski). In the realm of fuzzy probability theory, statistical maps describe the transportation of probability measures on one measurable space into probability measures on another measurable space. We show that for discrete probability spaces each statistical map can be represented via a special matrix the rows of which are probability functions related to conditional probabilities and the columns are related to fuzzy n-partitions of the domain. Discrete statistical maps sending a probability measure p to a probability measure q can be represented via conditional distributions and correspond to joint probabilities on the product. The composition of statistical maps provide a tool to describe and to study generalized random walks and Markov chains.

Design and research of a novel type of mechanical presses driven by three servo motors in parallel

2012 ◽  
Vol 227 (3) ◽  
pp. 580-591 ◽  
Author(s):  
Xun Huang ◽  
Chunxiang Ma ◽  
Suxing Wu ◽  
Qian Wang ◽  
Zhibo Wang ◽  
...  
Keyword(s):  
Probability Theory ◽  
Forward Kinematics ◽  
Generalized Function ◽  
Engineering Practice ◽  
Drawing Process ◽  
Fuzzy Probability ◽  
Fuzzy Reliability ◽  
Deep Drawing Process ◽  
Computing Method ◽  
Fuzzy Probability Theory

A novel type of mechanical presses design driven by three servo motors in parallel is proposed to develop high stamping capacity based on the theory of generalized function sets, and then is improved to enhance its practicability in engineering practice. The inverse and forward kinematics analyses are conducted later. Based on the fuzzy probability theory, the computing method of fuzzy reliability of kinematic accuracy of slider is then obtained. Finally, with a set of optimized length parameters of the presses given, the fuzzy reliability is calculated, on the basis of which a typical deep-drawing process is simulated.

Study on Fuzzy Reliability of Shaft Fixing Grinding Wheel in Internal Grinding

2006 ◽  
Vol 304-305 ◽  
pp. 218-221
Author(s):  
Chun Xiang Ma ◽  
Z.H. Qu ◽  
Q.J. Wu ◽  
Zhen Ruan ◽  
Li Ming Xu
Keyword(s):  
Probability Theory ◽  
Fatigue Strength ◽  
Set Theory ◽  
Fuzzy Set Theory ◽  
Grinding Wheel ◽  
Reliability Model ◽  
Fuzzy Probability ◽  
Fuzzy Reliability ◽  
Internal Grinding ◽  
Fuzzy Probability Theory

The fuzzy allowable intervals of the deflection, the vibration frequency and the fatigue strength of the shaft fixing grinding wheel are proposed in present paper based fuzzy set theory. According to fuzzy probability theory, the fuzzy reliability model of the rigidity, the fuzzy reliability model of the vibration and the fuzzy reliability model of the fatigue for the shaft fixing grinding wheel in internal grinding are given. The method is put into real example.

scholarly journals Upgrading Probability via Fractions of Events

2016 ◽  
Vol 24 (1) ◽  
pp. 29-41 ◽  
Author(s):  
Roman Frič ◽  
Martin Papčo
Keyword(s):  
Probability Theory ◽  
Random Variables ◽  
Random Variable ◽  
Classical Probability ◽  
Classical Probability Theory ◽  
Black And White ◽  
Random Events ◽  
Indicator Functions ◽  
Quantum Phenomena ◽  
Special Case

Abstract The influence of “Grundbegriffe” by A. N. Kolmogorov (published in 1933) on education in the area of probability and its impact on research in stochastics cannot be overestimated. We would like to point out three aspects of the classical probability theory “calling for” an upgrade: (i) classical random events are black-and-white (Boolean); (ii) classical random variables do not model quantum phenomena; (iii) basic maps (probability measures and observables { dual maps to random variables) have very different “mathematical nature”. Accordingly, we propose an upgraded probability theory based on Łukasiewicz operations (multivalued logic) on events, elementary category theory, and covering the classical probability theory as a special case. The upgrade can be compared to replacing calculations with integers by calculations with rational (and real) numbers. Namely, to avoid the three objections, we embed the classical (Boolean) random events (represented by the f0; 1g-valued indicator functions of sets) into upgraded random events (represented by measurable {0; 1}-valued functions), the minimal domain of probability containing “fractions” of classical random events, and we upgrade the notions of probability measure and random variable.

scholarly journals Probabilistic Investigations on the Explosion of Solutions of the KAC Equation with Infinite Energy Initial Distribution

2008 ◽  
Vol 45 (1) ◽  
pp. 95-106 ◽  
Author(s):  
Eric Carlen ◽  
Ester Gabetta ◽  
Eugenio Regazzini
Keyword(s):  
Random Variables ◽  
Random Variable ◽  
Initial Distribution ◽  
Initial Energy ◽  
Infinite Variance ◽  
Borel Sets ◽  
Second Moment ◽  
Quantitative Estimates ◽  
Kac Equation ◽  
Kac’S Model

Gabetta and Regazzini (2006b) have shown that finiteness of the initial energy (second moment) is necessary and sufficient for the solution of the Kac's model Boltzmann equation to converge weakly (Cb-convergence) to a probability measure on R. Here, we complement this result by providing a detailed analysis of what does actually happen when the initial energy is infinite. In particular, we prove that such a solution converges vaguely (C0-convergence) to the zero measure (which is identically 0 on the Borel sets of R). More precisely, we prove that the total mass of the limiting distribution splits into two equal masses (of value ½ each), and we provide quantitative estimates on the rate at which such a phenomenon takes place. The methods employed in the proofs also apply in the context of sums of weighted independent and identically distributed random variables x̃1, x̃2, …, where these random variables have an infinite second moment and zero mean. Then, with Tn := ∑j=1ηnλj,nx̃j, with max1 ≤ j ≤ ηnλj,n → 0 (as n → +∞), and ∑j=1ηnλj,n2 = 1, n = 1, 2, …, the classical central limit theorem suggests that T should in some sense converge to a ‘normal random variable of infinite variance’. Again, in this setting we prove quantitative estimates on the rate at which the mass splits into adherent masses to -∞ and +∞, or to ∞, that are analogous to those we have obtained for the Kac equation. Although the setting in this case is quite classical, we have not uncovered any previous results of a similar type.

scholarly journals Real Functions and the Extension of Generalized Probability Measures

2013 ◽  
Vol 55 (1) ◽  
pp. 85-94
Author(s):  
Jana Havlíčková
Keyword(s):  
Category Theory ◽  
Continuous Functions ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Classical Probability ◽  
Random Events ◽  
Elementary Methods ◽  
Generalized Probability ◽  
Fuzzy Probability Theory

Abstract In the classical probability, as well as in the fuzzy probability theory, random events and probability measures are modelled by functions into the closed unit interval [0,1]. Using elementary methods of category theory, we present a classification of the extensions of generalized probability measures (probability measures and integrals with respect to probability measures) from a suitable class of generalized random events to a larger class having some additional (algebraic and/or topological) properties. The classification puts into a perspective the classical and some recent constructions related to the extension of sequentially continuous functions.

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