Probability Theory

Statistics deals with the collection and interpretation of data. This chapter lays a foundation that allows to rigorously describe non-deterministic processes and to reason about non-deterministic quantities. The mathematical framework is given by probability theory, whose objects of interest are random quantities, their description and properties.

Hospital Practice, 1990

African Journal of Mathematics and Computer Science Research, 2010

Probability is the measure of the degree of confidence with which a certain event is expected to happen, among those possible in that context and in that situation.

Abdella Mohammed Ahmed (Msc) , 2024

The concept of a probability space that completely describes the outcome of a random experiment has been developed in chapter II. In this chapter, we develop the idea of a function defined on the outcome of a random experiment, which is a very high- level definition of a random variable. Thus, the value of a random variable is a random phenomenon and is a numerically valued random phenomenon. A random variable could be discrete or continuous. In this chapter, we will see probability distribution of discrete random variable and probability density function of continuous random variable. Finally, we will try to see the cumulative probability distribution of discrete and continuous variable and the expected values, variance and standard deviations of discrete and continuous random variable.

Mathematical Structures in Computer Science, 2014

In this paper, we discuss the crucial but little-known fact that, as Kolmogorov himself claimed, the mathematical theory of probabilities cannot be applied to factual probabilistic situations. This is because it is nowhere specified how, for any given particular random phenomenon, we should construct, effectively and without circularity, the specific and stable distribution law that gives the individual numerical probabilities for the set of possible outcomes. Furthermore, we do not even know what significance we should attach to the simple assertion that such a distribution law “exists”. We call this problem Kolmogorov's aporia†.We provide a solution to this aporia in this paper. To do this, we first propose a general interpretation of the concept of probability on the basis of an example, and then develop it into a non-circular and effective general algorithm of semantic integration for the factual probability law involved in a specific factual probabilistic situation. The dev...

2012

While mathematicians would define thinking probabilistically in terms of adequate use of probabilistic models, individuals are faced with the context of the situations to be modelled. And the ingredients of these situations could lead to directions completely different from standard mathematical models and their solutions. For didactical purpose it has to be clarified in which respects thinking probabilistically could be characterized. Some features are illustrated by context and comprise amongst others:

EOLSS Publishers

The philosophy of probability is a well-established, yet still greatly expanding field within the philosophy of science, which focuses upon questions regarding the nature and interpretation of the notion of probability; the connections between probability and metaphysical chance; and the role that the notion of probability plays in statistical modelling practice across the sciences. This chapter provides a state of the art review of the philosophy of probability with an eye on the fundamental duality and irreducible pluralism of objective and subjective variants of chance. Throughout the essay I emphasize the different ways in which an appropriate articulation of the subjective dimension of probability has historically been facilitated by a proper regard for its objective dimension of proba bility, and viceversa.

In this article, we are interested in an introductory teaching of the probabilistic formalism at university level, in particular around the notion of random variable. Our research hypothesis is that a teaching based on a formal approach, even if it is intended for second year students of the Bachelor of Science degree, can be doomed to a didactic failure. Our study with a small number of students, but over a long duration of observations, has allowed to raise various conceptual difficulties and obstacles around the definition and production of random variable examples. The difficulties that impede the availability of this object are mainly due to conceptual confusions between the concept of random variable and the notions of image universe, random experiment, or law of probabilities. A quantitative analysis of the productions of students showed that the relevance of the formal approach was without effect on the production of example, whereas that of the intuitive approach had an effect on the validity of the production of example of random variable. These results encourage thus the adoption of a dialectical formalism/intuition in the introduction of the probabilistic formalism; such an approach of teaching would seem to be a priori quite in favor of a good apprehension by the students.