Diagrams for difficult problems in probability

2003 ◽  
Vol 87 (508) ◽  
pp. 86-97 ◽  
Author(s):  
Peter C.-H. Cheng ◽  
Nigel G. Pitt
Keyword(s):  
Probability Theory ◽  
Conceptual Understanding ◽  
Probability Space ◽  
Learning And Instruction

We have developed a novel diagrammatic approach for understanding and teaching probability theory — Probability Space diagrams [1]. Our studies of learning and instruction with Probability Space (PS) diagrams have demonstrated that they can significantly enhance students' conceptual understanding. This article illustrates the utility of PS diagrams by applying them to the explanation of some difficult concepts and notoriously counterintuitive problems in probability. We first outline the nature of the system.

Probability in quantum mechanics

1978 ◽  
Vol 10 (4) ◽  
pp. 725-729 ◽  
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

scholarly journals Forming a conceptual understanding of Mathematics at students of technical universities

2020 ◽  
pp. 77-90
Author(s):  
Elena V. Kuznetsova ◽  
◽  
Natalia Yu. Zhbanova
Keyword(s):  
Higher Education ◽  
Probability Theory ◽  
University Students ◽  
Computer Modeling ◽  
Conceptual Understanding ◽  
Effective Means ◽  
Educational Process ◽  
Point Of View ◽  
Specific Nature ◽  
Global Challenges

The global challenges that humanity is facing today pose the task of higher education to prepare a specialist with fundamental training and the ability to learn throughout life. Fundamentalization of education is not possible without the formation of students' conceptual understanding of the material studied. This problem is quite relevant in the study of mathematics due to the specific nature of this science. The researchers' lack of a unified point of view in determining the essence of the conceptual understanding of mathematics does not allow practitioners to develop tools for assessing the level of conceptual understanding among university students. The purpose the article is to identify and formulate the essential characteristics and pedagogical conditions for the formation of a conceptual understanding of mathematics, as well as to explore the possibilities and effectiveness of the integration of computer modeling as an instrument of forming a conceptual understanding in the process of teaching probability theory to students of technical universities. The study showed that a computer workshop, developed based on identified pedagogical conditions and taking into account the didactic capabilities of ICT in the educational process, is an effective means of developing conceptual understanding when studying a course in probability theory. Students whose curriculum included a workshop with elements of computer modeling have a more excellent knowledge of methodologically significant knowledge and the ability to relate previously learned material to new problems.

Probability in quantum mechanics

1978 ◽  
Vol 10 (04) ◽  
pp. 725-729
Author(s):  
J. V. Corbett
Keyword(s):  
Probability Theory ◽  
Hilbert Space ◽  
Quantum Mechanics ◽  
Mechanical System ◽  
Partial Order ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Mathematical Expression ◽  
Quantum Mechanical ◽  
Quantum Mechanical System

Quantum mechanics is usually described in the terminology of probability theory even though the properties of the probability spaces associated with it are fundamentally different from the standard ones of probability theory. For example, Kolmogorov's axioms are not general enough to encompass the non-commutative situations that arise in quantum theory. There have been many attempts to generalise these axioms to meet the needs of quantum mechanics. The focus of these attempts has been the observation, first made by Birkhoff and von Neumann (1936), that the propositions associated with a quantum-mechanical system do not form a Boolean σ-algebra. There is almost universal agreement that the probability space associated with a quantum-mechanical system is given by the set of subspaces of a separable Hilbert space, but there is disagreement over the algebraic structure that this set represents. In the most popular model for the probability space of quantum mechanics the propositions are assumed to form an orthocomplemented lattice (Mackey (1963), Jauch (1968)). The fundamental concept here is that of a partial order, that is a binary relation that is reflexive and transitive but not symmetric. The partial order is interpreted as embodying the logical concept of implication in the set of propositions associated with the physical system. Although this model provides an acceptable mathematical expression of the probabilistic structure of quantum mechanics in that the subspaces of a separable Hilbert space give a representation of an ortho-complemented lattice, it has several deficiencies which will be discussed later.

On probabilistic argumentation and subargument-completeness

2021 ◽  
Author(s):  
Régis Riveret ◽  
Nir Oren
Keyword(s):  
Probability Theory ◽  
Probability Space ◽  
Short Communication ◽  
Sample Space ◽  
Formal Models ◽  
Probabilistic Argumentation

Abstract Probabilistic argumentation combines probability theory and formal models of argumentation. Given an argumentation graph where vertices are arguments and edges are attacks or supports between arguments, the approach of probabilistic labellings relies on a probability space where the sample space is any specific set of argument labellings of the graph, so that any labelling outcome can be associated with a probability value. Argument labellings can feature a label indicating that an argument is not expressed, and in previous work these labellings were constructed by exploiting the subargument-completeness postulate according to which if an argument is expressed then its subarguments are expressed and through the use of the concept of ‘subargument-complete subgraphs’. While the use of such subgraphs is interesting to compare probabilistic labellings with other works in the literature, it may also hinder the comprehension of a relatively simple framework. In this short communication, we revisit the construction of probabilistic labellings and demonstrate how labellings can be specified without reference to the concept of subargument-complete subgraphs. By doing so, the framework is simplified and yields a more natural model of argumentation.

Fuzzy Quantum Space

2017 ◽  
pp. 84-114
Author(s):  
Renáta Bartková ◽  
Beloslav Riečan ◽  
Anna Tirpáková
Keyword(s):  
Probability Theory ◽  
Boolean Algebra ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Probability Space ◽  
Quantum Space ◽  
Fuzzy State ◽  
Fuzzy Observable

In this chapter we study the existence of a sum of fuzzy observables in a fuzzy quantum space which generalizes the Kolmogorov probability space using the ideas of fuzzy set theory. We also study some properties of the sum of fuzzy observables. To study the above mentioned, we also include the basic notions from the probability theory on fuzzy quantum space in this chapter, i.e. the notion of fuzzy quantum space, a fuzzy observable, an indicator of a fuzzy set, a null fuzzy observable, a Boolean algebra on fuzzy quantum space, fuzzy state etc.

scholarly journals A probability space of continuous-time discrete value stochastic process with Markov property

2016 ◽  
Vol 12 (3) ◽  
pp. 5975-5991
Author(s):  
Miloslawa Sokol
Keyword(s):  
Stochastic Process ◽  
Probability Theory ◽  
Stochastic Processes ◽  
Continuous Time ◽  
Probability Space ◽  
Measure Theory ◽  
Markov Property ◽  
Finite Measure ◽  
Physical Models

Getting acquainted with the theory of stochastic processes we can read the following statement: "In the ordinary axiomatization of probability theory by means of measure theory, the problem is to construct a sigma-algebra of measurable subsets of the space of all functions, and then put a finite measure on it". The classical results for limited stochastic and intensity matrices goes back to Kolmogorov at least late 40-s. But for some infinity matrices the sum of probabilities of all trajectories is less than 1. Some years ago I constructed physical models of simulation of any stochastic processes having a stochastic or an intensity matrices and I programmed it. But for computers I had to do some limitations - set of states at present time had to be limited, at next time - not necessarily. If during simulation a realisation accepted a state out of the set of limited states - the simulation was interrupted. I saw that I used non-quadratic, half-infinity stochastic and intensity matrices and that the set of trajectories was bigger than for quadratic ones. My programs worked good also for stochastic processes described in literature as without probability space. I asked myself: did the probability space for these experiments not exist or were only set of events incompleted? This paper shows that the second hipothesis is true.

On Multiple Integral Geometric Integrals and Their Applications to Probability Theory

1970 ◽  
Vol 22 (1) ◽  
pp. 151-163 ◽  
Author(s):  
Franz Streit
Keyword(s):  
Probability Theory ◽  
Euclidean Space ◽  
Integral Geometry ◽  
Mathematical Theory ◽  
Probability Space ◽  
Convex Bodies ◽  
Multiple Integral ◽  
Mean Value ◽  
Dimensional Euclidean Space ◽  
Point Sets

It has been pointed out repeatedly in the literature that the methods of integral geometry (a mathematical theory founded by Wilhelm Blaschke and considerably extended by several mathematicians) provide highly suitable means for the solution of problems concerning “geometrical probabilities“ [2; 6; 12; 15]. The possibilities for the application of these integral geometric results to the evaluation of probabilities, satisfying certain conditions of invariance with respect to a group of transformations which acts on the probability space, are obviously not yet exhausted. In this article, such applications are presented. First, some concepts and notation are introduced (§1). In the next section we derive some integral geometric relations (§ 2). These results are generalizations of known systems of formulae and they are valid in the k-dimensional Euclidean space. In § 3, we determine mean-value formulae for the fundamental characteristics of point-sets, generated by randomly placed convex bodies.

Stochastic independence in non-commutative probability theory

1979 ◽  
Vol 86 (1) ◽  
pp. 103-114
Author(s):  
Wulf Driessler ◽  
Ivan F. Wilde
Keyword(s):  
Probability Theory ◽  
Probability Space ◽  
Random Variables ◽  
Independent Random Variables ◽  
Characteristic Functions ◽  
Usual Sense ◽  
Stochastic Independence ◽  
Selfadjoint Operators

AbstractFor a family {Xα} of random variables over a probability space , stochastic independence can be formulated in terms of factorization properties of characteristic functions. This idea is reformulated for a family {Aα} of selfadjoint operators over a probability gage space and is shown to be inappropriate as a non-commutative generalization. Indeed, such factorization properties imply that the {Aα} mutually commute and are versions of independent random variables in the usual sense.

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