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In probability theory, a probability space or a probability triple (,,) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: [1] [2]
A probability space is a measure space with a probability measure. For measure spaces that are also topological spaces various compatibility conditions can be placed for the measure and the topology. Most measures met in practice in analysis (and in many cases also in probability theory) are Radon measures.
A stochastic process is defined as a collection of random variables defined on a common probability space (,,), where is a sample space, is a -algebra, and is a probability measure; and the random variables, indexed by some set , all take values in the same mathematical space , which must be measurable with respect to some -algebra .
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
A random variable is a measurable function: from a sample space as a set of possible outcomes to a measurable space.The technical axiomatic definition requires the sample space to be a sample space of a probability triple (,,) (see the measure-theoretic definition).
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
A well-defined, non-empty sample space is one of three components in a probabilistic model (a probability space). The other two basic elements are a well-defined set of possible events (an event space), which is typically the power set of S {\displaystyle S} if S {\displaystyle S} is discrete or a σ-algebra on S {\displaystyle S} if it is ...
A measurable subset of a standard probability space is a standard probability space. It is assumed that the set is not a null set, and is endowed with the conditional measure. See (Rokhlin 1952, Sect. 2.3 (p. 14)) and (Haezendonck 1973, Proposition 5). Every probability measure on a standard Borel space turns it into a standard probability space.
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