Basic Probability Theory and Random Numbers

2001 ◽  
pp. 197-211
Author(s):  
José M. Garrido
Keyword(s):  
Probability Theory ◽  
Random Numbers ◽  
Basic Probability

Basic probability theory

2019 ◽  
pp. 121-140
Author(s):  
Thomas P. Trappenberg
Keyword(s):  
Machine Learning ◽  
Probability Theory ◽  
Random Numbers ◽  
Density Functions ◽  
Approximation Techniques ◽  
Basic Probability ◽  
Probability Mass ◽  
Learning Probability ◽  
Mass Functions ◽  
Modern Machine

The discussion provides a refresher of probability theory, in particular with respect to the formulations that build the theoretical language of modern machine learning. Probability theory is the formalism of random numbers, and this chapter outlines what these are and how they are characterized by probability density or probability mass functions. How such functions have traditionally been characterized is covered, and a review of how to work with such mathematical objects such as transforming density functions and how to measure differences between density function is presented. Definitions and basic operations with multiple random variables, including the Bayes law, are covered. The chapter ends with an outline of some important approximation techniques of so-called Monte Carlo methods.

scholarly journals On Basic Probability Logic Inequalities

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1409
Author(s):  
Marija Boričić Joksimović
Keyword(s):  
Probability Theory ◽  
Inference Rule ◽  
Modus Ponens ◽  
Probability Logic ◽  
Modus Tollens ◽  
Short Overview ◽  
Basic Probability ◽  
Probabilistic Version ◽  
Hypothetical Syllogism

We give some simple examples of applying some of the well-known elementary probability theory inequalities and properties in the field of logical argumentation. A probabilistic version of the hypothetical syllogism inference rule is as follows: if propositions A, B, C, A→B, and B→C have probabilities a, b, c, r, and s, respectively, then for probability p of A→C, we have f(a,b,c,r,s)≤p≤g(a,b,c,r,s), for some functions f and g of given parameters. In this paper, after a short overview of known rules related to conjunction and disjunction, we proposed some probabilized forms of the hypothetical syllogism inference rule, with the best possible bounds for the probability of conclusion, covering simultaneously the probabilistic versions of both modus ponens and modus tollens rules, as already considered by Suppes, Hailperin, and Wagner.

Basic probability theory

1992 ◽  
pp. 21-53
Author(s):  
Roy Billinton ◽  
Ronald N. Allan
Keyword(s):  
Probability Theory ◽  
Basic Probability

scholarly journals Geometric interpretation of subjective probability: random numbers and objective conditions of coherence

2016 ◽  
Vol 4 (2) ◽  
pp. 90
Author(s):  
Pierpaolo Angelini ◽  
Antonio Maturo
Keyword(s):  
Probability Theory ◽  
Coordinate System ◽  
Subjective Probability ◽  
Geometric Interpretation ◽  
Psychological Aspect ◽  
Random Numbers ◽  
Metric Properties ◽  
Probabilistic Evaluation ◽  
The Individual

In the domain of the logic of certainty we study the objective notions of the subjective probability with the clear aim of identifying their fundamental characteristics before the assignment, by the individual, of the probabilistic evaluation: probability is an additional and subjective notion that one applies within the range of possibility, thus giving rise to those gradations, more or less probable, that are meaningless in the logic of certainty. When we study the criteria for evaluations under conditions of uncertainty and their corresponding conditions of coherence we show an inevitable dichotomy between the subjective or psychological aspect of probability and the objective or logical or geometrical one. The affine properties are the basis of essential concepts of probability theory and only they make sense, being independent of the choice of a coordinate system; however, the importance of the metric properties appears in order to represent random numbers and analytical conditions of coherence.

Basic probability theory

1983 ◽  
pp. 5-35
Author(s):  
Roy Billinton ◽  
Ronald N. Allan
Keyword(s):  
Probability Theory ◽  
Basic Probability
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