Probability Space

In probability theory, a probability space or a probability triple is a mathematical construct that models a real-world process (or "experiment") consisting of states that occur randomly. A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probability levels are also the same.

A probability space consists of three parts:

  1. A sample space, Ω, which is the set of all possible outcomes.
  2. A set of events, where each event is a set containing zero or more outcomes.
  3. The assignment of probabilities to the events, that is, a function P from events to probability levels.

An outcome is the result of a single execution of the model. Since individual outcomes might be of little practical use, more complex events are used to characterize groups of outcomes. The collection of all such events is a σ-algebra . Finally, there is a need to specify each event's likelihood of happening. This is done using the probability measure function, P.

Once the probability space is established, it is assumed that “nature” makes its move and selects a single outcome, ω, from the sample space Ω. All the events in that contain the selected outcome ω (recall that each event is a subset of Ω) are said to “have occurred”. The selection performed by nature is done in such a way that if the experiment were to be repeated an infinite number of times, the relative frequencies of occurrence of each of the events would coincide with the probabilities prescribed by the function P.

The prominent Soviet mathematician Andrey Kolmogorov introduced the notion of probability space, together with other axioms of probability, in the 1930s. Nowadays alternative approaches for axiomatization of probability theory exist; see “Algebra of random variables”, for example.

This article is concerned with the mathematics of manipulating probabilities. The article probability interpretations outlines several alternative views of what "probability" means and how it should be interpreted. In addition, there have been attempts to construct theories for quantities that are notionally similar to probabilities but do not obey all their rules; see, for example, Free probability, Fuzzy logic, Possibility theory, Negative probability and Quantum probability.

Read more about Probability Space:  Introduction, Definition, Discrete Case, General Case, Non-atomic Case, Complete Probability Space

Other articles related to "probability, probability space, space, probability spaces":

Multiple-try Metropolis
... the Metropolis–Hastings algorithm (MH) can be used to sample from a probability distribution which is difficult to sample from directly ... one uses a Gaussian distribution centered on the current point in the probability space, of the form ... accepted, and the Markov chain will be similar to a random walk through the probability space ...
Kolmogorov Extension Theorem - Implications of The Theorem
... The measure-theoretic approach to stochastic processes starts with a probability space and defines a stochastic process as a family of functions on this probability space ... requirements, one can always identify a probability space to match the purpose ... does not have to be explicit about what the probability space is ...
Probability Space - Related Concepts - Mutual Exclusivity
... However, the probability of the union of an uncountable set of events is not the sum of their probabilities ...
Standard Probability Space
... In probability theory, a standard probability space, also called Lebesgue–Rokhlin probability space or just Lebesgue space (the latter term is ambiguous) is a probability space satisfying certain ... unit interval endowed with the Lebesgue measure has important advantages over general probability spaces, and can be used as a probability space for all practical ... The theory of standard probability spaces was started by von Neumann in 1932 and shaped by Vladimir Rokhlin in 1940 ...
Stochastic Differential Equation - Use in Probability and Mathematical Finance
... The notation used in probability theory (and in many applications of probability theory, for instance mathematical finance) is slightly different ... The difference between the two lies in the underlying probability space (Ω, F, Pr) ... A weak solution consists of a probability space and a process that satisfies the integral equation, while a strong solution is a process that satisfies the equation and is defined on a given probability space ...

Famous quotes containing the words space and/or probability:

    In the tale proper—where there is no space for development of character or for great profusion and variety of incident—mere construction is, of course, far more imperatively demanded than in the novel.
    Edgar Allan Poe (1809–1849)

    Only in Britain could it be thought a defect to be “too clever by half.” The probability is that too many people are too stupid by three-quarters.
    John Major (b. 1943)