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Probability theory or probability calculus is the branch of mathematics concerned with probability. Although there are several different probability interpretations , probability theory treats the concept in a rigorous mathematical manner by expressing it through a set of axioms .
A Treatise on Probability, [1] published by John Maynard Keynes in 1921, provides a much more general logic of uncertainty than the more familiar and straightforward 'classical' theories of probability. [notes 1] [3] [notes 2] This has since become known as a "logical-relationist" approach, [5] [notes 3] and become regarded as the seminal and ...
The Wiener process is widely considered the most studied and central stochastic process in probability theory. [1] [2] [3] In probability theory and related fields, a stochastic (/ s t ə ˈ k æ s t ɪ k /) or random process is a mathematical object usually defined as a family of random variables in a probability space, where the index of the ...
The purely mathematical analysis of random variables is independent of such interpretational difficulties, and can be based upon a rigorous axiomatic setup. In the formal mathematical language of measure theory, a random variable is defined as a measurable function from a probability measure space (called the sample space) to a measurable space.
Traditional probability theory does not state if a specific sequence is random, but generally proceeds to discuss the properties of random variables and stochastic sequences assuming some definition of randomness. The Bourbaki school considered the statement "let us consider a random sequence" an abuse of language. [3]
A random process is a sequence of random variables whose outcomes do not follow a deterministic pattern, but follow an evolution described by probability distributions. These and other constructs are extremely useful in probability theory and the various applications of randomness .
In probability theory, a random measure is a measure-valued random element. [ 1 ] [ 2 ] Random measures are for example used in the theory of random processes , where they form many important point processes such as Poisson point processes and Cox processes .
A Philosophical Essay on Probabilities is a work by Pierre-Simon Laplace on the mathematical theory of probability. [1] [2] [3] The book consists of two parts, the first with five chapters and the second with thirteen. [1]