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Introduction to Logic and Probability
Published in Sriraman Sridharan, R. Balakrishnan, Foundations of Discrete Mathematics with Algorithms and Programming, 2019
Sriraman Sridharan, R. Balakrishnan
We define the finite probability space (probability model) as follows: A probability model (probability space) is simply a sample space with a non-negative real number associated with each point of the sample space with the condition that the sum of all the numbers associated with the points of the sample space is 1. The non-negative number associated with a point is called the probability of that point. We denote by p(s), the probability of the point s in S. The probability of an event E⊂S $ E\subset S $ is denoted by p(E) and is the sum of all the probabilities of the points of E, that is, p(E)=∑einEp(e) $ p(E)=\sum _{ e in E} p(e) $ .
Probability and Statistics
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
Technically, a probability space consists of three parts: a sample space S (set of possible outcomes); a set of events {A} (an event contains a set of outcomes); and a probability measure P (a function on events such that P(A) ≥ 0, P(S) = 1, and P(∪j∈JAj)=∑j∈JP(Aj) $ P( \cup_{j \in J} A_{j} ) = \mathop {\sum_{j \in J} }\limits_{{}}^{{}} P(A_{j} ) $ if {Aj|j ∊ J} is a countable, pairwise disjoint collection of events).
Formal Reliability Analysis of Railway Systems Using Theorem Proving Technique
Published in Qamar Mahboob, Enrico Zio, Handbook of RAMS in Railway Systems, 2018
Waqar Ahmad, Osman Hasan, Sofiène Tahar
Mathematically, a measure space is defined as a triple (Ω, Σ, µ), where Ω is a set, called the sample space; Σ represents a σ algebra of subsets of Ω, where the subsets are usually referred to as measurable sets; and µ is a measure with domain Σ. A probability space is a measure space (Ω, Σ, Pr), such that the measure, referred to as the probability and denoted by Pr, of the sample space is 1. In the HOL formalization of probability theory [10], given a probability space p, the functions , , and return the corresponding Ω, Σ, and Pr, respectively. This formalization also includes the formal verification of some of the most widely used probability axioms, which play a pivotal role in formal reasoning about reliability properties.
Independent events and their complements
Published in International Journal of Mathematical Education in Science and Technology, 2021
Carolina Martins Crispim, Gabriel Perez Mizuno, Adrian Pizzinga
By a probability space, we mean any triple , where Ω is a nonempty set called the sample space, is an appropriate σ-field2 of the subsets of Ω and P is a probability measure defined on . The usual introductory motivations and terminology are pertinent here: the points in Ω are interpreted as the possible outcomes of an experiment; the subsets of Ω may be called events – in particular, the events belonging to , the ones that have probabilities, are called random events; P reflects the mathematical perception of chance and, in practical problems, is frequently achieved after a proper statistical data analysis; and so on. Here, intuition is welcome. Even more, it is encouraged.
Adaptive sliding mode control for interval type-2 stochastic fuzzy systems subject to actuator failures
Published in International Journal of Systems Science, 2018
Notations: In this paper, is the expectation operator and denotes the Euclidean norm. For a real symmetric matrix, is used to represent that P is a positive definite (or positive semidefinite) matrix, and means . I denotes an identity matrix with the proper dimension. The transpose of the matrix A is denoted by and tr A is the trace of A. In symmetric matrices, is an ellipsis for a block matrix that can be induced by symmetry. represents the probability space with Ω the sample space, the σ-algebra of subsets of the sample space and the probability measure on . Matrices, if not explicitly stated, are assumed to have compatible dimensions.